European Mathematical Society - 15 Linear and multilinear algebra, matrix theory
https://euro-math-soc.eu/msc/15-linear-and-multilinear-algebra-matrix-theory
en Linear Algebra and Optimization with Applications to Machine Learning. Volume I: Linear Algebra for Computer Vision, Robotics, and Machine Learning
https://euro-math-soc.eu/review/linear-algebra-and-optimization-applications-machine-learning-volume-i-linear-algebra
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>The "with applications" in the title of the book should be read as "applicable" because it provides the fundamental mathematics, but it does not explicitly treat the applications that are mentioned. In this volume I, these fundamentals are basically linear algebra, and in volume II it is promised to cover optimization. I can imagine though that there is some interaction like for example the important subject of linear programming. Except for a few illustrative examples, the applications themselves are supposed to be covered in other courses or textbooks. There is a bit of statistics, which I think is also important for the applications mentioned, but probability and statistics as well as calculus is assumed known, but anyway, whatever is needed from these is recalled briefly.</p>
<p>Since this volume introduces the fundamentals with applications in mind, it is in some sense similar to a first course in linear algebra that I have been teaching to engineering students for many years. This volume has also some numerical procedures and even matlab codes. Those I taught in a separate numerical course. The procedures discussed include Gaussian elimination, Cholesky factorization, QR decomposition, eigenvalue and SVD algorithms, Krylov and Lanczos methods, but it skips the basic numerics of rounding errors, error propagation, numerical stability, and the more analytic problems such as numerical quadrature, differential equations, zero finding, etc. The latter also have a definite link with linear algebra, but clearly including all applications of linear algebra is an interminable task.</p>
<p>I wrote my own lecture notes, not satisfied with the existing books that were not providing the desired abstraction and that spent too many glossy pages on the introductory level with many examples and applications. In many ways my notes were very similar to the material covered here, which is why I like this book so much. But since I was trying to cover as much as possible in the most efficient way restricted by the amount of credits that were assigned to the course, my notes were much more concise. This is quite different in this book since, at its introductory level, it is almost encyclopedic and it is like the text I would have liked to write if I were not restricted by a time limit to cover all the material. For example, I spent some time explaining that finite dimensional real vector spaces and linear maps can be treated in an isomorphic way by discussing $\mathbb{R}^n$ and matrix algebra, and then I could just do matrix algebra. Not so in this book. The abstract vector spaces remain present throughout the book. Infinite dimensional vector spaces are a problem because there you need infinite sums and convergence, which require topology to define convergence, and the maps are operators. The authors here maintain some elements of function spaces but certain analysis aspects are not really covered in detail. But otherwise almost all the proofs are fully written out.</p>
<p>Thanks to LaTeX it is nowadays no problem to produce a professionally looking text. The illustrations however require different tools and producing good quality graphics is a challenge. Clearly the authors of this book had the same problem with graphics that I also had. The text is excellent, but the graphics are definitely of lesser quality. Unfortunately it is not only a problem of how they are generated and reproduced, also they do not always make very clear what they are supposed to illustrate.</p>
<p>What would one expect in a basic linear algebra course for engineering-type students? I think that should include vector spaces and linear maps and how they relate to matrices, the rank of a matrix with range and null space, determinants (in my opinion as little as possible), linear systems with Gaussian elimination, normed and Euclidean spaces, orthogonalization and QR, eigenvalue and singular values with generalized inverses and least squares, and geometric interpretation of all these concepts. All this is extensively discussed in this book. The numerical and matlab algorithms and certainly the iterative methods, I would rather expect in a more specialized numerical course, but it is not completely unexpected that they are found here in this book. More unexpected are the following: A discussion about the Haar wavelet with some signal and image processing; the chapter on linear systems is introduced with a discussion about the computation of interpolating Bézier curves; and the computation of the matrix exponential, important for the solution of differential equations, is discussed to some extent. There is also an extensive discussion of groups (SU(2), SO(3), and quaternions. This is important for robotics. Finally, there is much material related to graphs: graph Laplacians, clusters, and graph drawing. The final chapter about polynomial factorization and the Jordan form is less elementary and not always found with the same detail in a basic course. So there is a lot of material, and I can imagine that one wants to make a selection. No advise is given by the authors and it would be difficult anyway since successive chapters rely on previous ones. The authors have earmarked only some sections that they considered to be more advanced and these can be skipped initially.</p>
<p>To conclude, I can definitely recommend this extensive book on linear algebra that is both self contained and thorough and mostly remaining at a basic level. I have always considered linear algebra a basic tool in many applications. The applications mentioned in the title are computer vision, robotics and machine learning but they are not really discussed. However the Haar wavelet is a hint to image and signal processing, robotics are related to the study of the quaternions and the rotation groups, the linear algebra and the graphs are useful for a lot of applications, thus also for machine learning. To come to the actual applications though will need extra material, continuing the basics given here, but also some extra material for example statistics, and optimization (the latter is promised in volume II). Every chapter has a list of exercises (no answers are provided). There is a bibliography which lists mostly books and an index (12 pages). With a book of this size, the index can never be extensive enough. I tried to look up some topics that were not listed. On the other hand, I noted separate entries for "Jordan block" and "Jordan blocks", which makes no sense of course. I know from experience that collecting an index in an artisanal way is, just like generating good graphics, a time consuming task that needs patience and many iterations.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This is a extensive and thorough introduction to linear algebra that includes some extras like wavelets, Bézier curves, groups of rotations, quaternions, and applications in graph theory, that are of particular interest for applications in computer vision, robotics, and machine learning.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/jean-gallier" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jean Gallier</a></li><li class="vocabulary-links field-item odd"><a href="/author/jocelyn-quaintance" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jocelyn Quaintance</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/world-scientific" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">world scientific</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2020</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9789811206399 (hbk), 9789811207716 (pbk), 9789811206412 (ebk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">GBP 175.00 (hbk), GBP 85.00 (pbk), GBP 70.00 (ebk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">824</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/algebra" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Algebra</a></li><li class="vocabulary-links field-item odd"><a href="/imu/numerical-analysis-and-scientific-computing" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Numerical Analysis and Scientific Computing</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://www.worldscientific.com/worldscibooks/10.1142/11446" title="Link to web page">https://www.worldscientific.com/worldscibooks/10.1142/11446</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/15-linear-and-multilinear-algebra-matrix-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">15 Linear and multilinear algebra, matrix theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/15axx" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">15Axx</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/65-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/65d19" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65D19</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/65t05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65T05</a></li></ul></span>Fri, 10 Apr 2020 06:07:24 +0000Adhemar Bultheel50668 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/linear-algebra-and-optimization-applications-machine-learning-volume-i-linear-algebra#commentsSeparable type representations of matrices and fast algorithms
https://euro-math-soc.eu/review/separable-type-representations-matrices-and-fast-algorithms
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Almost five years after Israel Gohberg passed away in 2009, his coauthors have finished this extensive round-up of their results on separable matrices. It has grown out to a thick manuscript that has been published as two volumes in the Birkhäuser series <em>Operator Theory: Advances and Applications</em>, Vols. 234, 235.</p>
<p>
It all started with the Toeplitz and Hankel matrices for which special algorithms existed in diverse branches of mathematics. The Levinson algorithm for Toeplitz matrices was published in 1945 and later connected with the Schur algorithm from 1917 and with Szego recurrence relation for orthogonal polynomials on the unit circle from 1921. A similar history is connected with Hankel matrices which can be connected with recurrence relations for orthogonal polynomials on the real line and with continued fractions. Systems with Hankel or Toeplitz matrices have the property that they can be stored in <em>O</em>(<em>n</em>) memory and solved in <em>O</em>(<em>n</em>²) computations rather than the usual <em>O</em>(<em>n</em>²) memory and <em>O</em>(<em>n</em>³) computations for general matrices.</p>
<p>
Around mid 1980s a new interest in such fast algorithms was enforced by the applications in signal processing and systems theory. Soon other matrices were discovered having a so-called low displacement rank, which could also be stored by some generators in <em>O</em>(<em>n</em>) memory and inverted in <em>O</em>(<em>n</em>²) computations. So the notion grew of matrices that were sparse not because they contain many zeros, but sparse because the information they store is sparse, although not by a visible pattern. This brought about a flourishing research area in linear algebra studying matrices that have some structured information that can be exploited for efficient storage and computation. The separable matrices as discussed in these books are such a class.</p>
<p>
The simplest form are the separable matrices. That is when <em>A</em> = <em>UV*</em> with <em>A</em> of size <em>N × N</em> but <em>U</em> and <em>V</em> of size <em>N × n</em> with <em>n ≪ N</em>. Only 2<em>nN</em> elements are needed to store it. Direct generalizations are semi-separable matrices whose lower and upper triangular parts are the lower and upper triangular parts of possibly different separable matrices. Also quasi-separable matrices are characterized by the maximal rank that a submatrix can have when all its elements are in the upper or lower triangular part of the matrix. There is often a relation with classical structures like being the inverse of a tridiagonal matrix, etc. Unitary upper or lower Hessenberg matrices is another class of structured matrices. The latter can be represented as a sequence of 2 by 2 Givens rotations applied to certain rows or columns. This is again a useful sparse representation. Some of these problems are related to minimal rank matrix completion problems: find the missing elements in a partially prescribed matrix that keeps the rank as small as possible.</p>
<p>
The authors have published about this subject during the last decades and their results are collected in these two volumes. In the first volume the basic elements are collected: definitions, fast matrix vector and matrix matrix multiplication, LDU, QR and other factorizations, inversion etc. for all kinds of structured matrices: separable, semiseparable, semiseparable-plus-diagonal, quasiseparable, Green matrices, unitary Hessenberg,... Much attention is given to obtaining the proper generators and to the completion problem.</p>
<p>
The second volume is reserved for eigenvalue problems. The structure of quasipseparable matrices, divide and conquer techniques, and QR based methods for Hermitean and unitary Hessenberg matrices, and for companion matrices as a special case.</p>
<p>
This is mainly a survey of the results of the authors. The two books are intimately coupled. It has no sense to start with volume 2 without having volume 1 available. The numbering of parts and chapters in volume 2 is continuing the numbering of volume 1. A similar two volume set was published before by R. Vandebril, M. Van Barel, and N. Mastronardi. <em>Matrix Computations and Semiseparable Matrices. Volume I: Linear Systems</em> and <em>Volume II: Eigenvalues and Singular Value Methods</em> (The John Hopkins University Press, Baltimore, 2008). Research is still extending the algebraic analysis as well as the numerical implementation of the algorithms for classes of matrices with structured information.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a compilation of the results obtained by the authors in the last decades about solving systems of equations and eigenvalue problems for matrices that have certain structural properties like separable, semi-separable, quasi-separable,semi-separable-plus-diagonal, unitary Hessenberg, etc.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/yuli-eidelman" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Yuli Eidelman</a></li><li class="vocabulary-links field-item odd"><a href="/author/israel-gohberg" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Israel Gohberg</a></li><li class="vocabulary-links field-item even"><a href="/author/iulian-haimovici" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Iulian Haimovici</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/birkh%C3%A4user-basel-0" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">birkhäuser, basel</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2014</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-3-0348-0728-9</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">158,99 €</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">788</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/algebra" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Algebra</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.springer.com/birkhauser/mathematics/book/978-3-0348-0728-9" title="Link to web page">http://www.springer.com/birkhauser/mathematics/book/978-3-0348-0728-9</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/15-linear-and-multilinear-algebra-matrix-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">15 Linear and multilinear algebra, matrix theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/15b99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">15B99</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/15a83" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">15A83</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/15a18" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">15a18</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/65f99" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65F99</a></li></ul></span>Thu, 17 Jul 2014 07:21:31 +0000Adhemar Bultheel45638 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/separable-type-representations-matrices-and-fast-algorithms#commentsEigenvalues of matrices / Françoise Chatelin, with exercises by Mario Ahués and Françoise Chatelin
https://euro-math-soc.eu/review/eigenvalues-matrices-fran%C3%A7oise-chatelin-exercises-mario-ahu%C3%A9s-and-fran%C3%A7oise-chatelin
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book was originally published in two separate volumes by Mason, Paris: "Valeurs propres de matrices" (1988) and "Exercises de valeurs propres de matrices" (1989). This SIAM edition is a revised republication of the first english version of the book, published by John Wiley & Sons, Inc., in 1993. </p>
<p>The present book is a work on numerical analysis in depth whose mail goal is to give a modern and complete theory, on an elementary level, of the problem of calculation of eigenvalues of square matrices. The first chapter is devoted to develop the theory of linear operators between finite dimensional complex vector spaces involved in the eigenvalue problem (and much more!). It is remarkable the strongly computational proof of the existence of the Jordan form of the endomorphisms of ${\mathbb C}^n$ based on the Schur's form. The collection of problems and exercises of this chapter is spectacular. </p>
<p>Along the book the author uses the language of functional analysis, which has the effect of demonstrating the profound similarity between the different methods of approximation, while the systematic use of bases to represent invariant subspaces provides a geometric interpretation that enhances the traditional algebraic presentation of many algorithms in numerical matrix analysis.</p>
<p>Elements of spectral theory are studied in Chapter 2. If ${\rm sp}(A)$ denotes the set of eigenvalues of $A$, the detailed study of the singularities of the analytic function<br />
$$<br />
{\mathbb C}\setminus{\rm sp}(A)\to{\mathfrak M}_n({\mathbb C}),\, z\mapsto(A-zI)^{-1},<br />
$$<br />
called the "resolvent" of $A$, give rise to the so called Rellich-Kato and Rayleigh-Schrdingen expansions.</p>
<p>In Chapter 3 the author explains why it is interesting to compute eigenvalues: differential and difference equations, Markov chains, theory of economics, factorial analysis of data, the dynamics of structures, chemistry and Fredholm's integral equations, are the selected topics where the knowledge of eigenvalues or their approximations are crucial.</p>
<p>Error analysis is treated in Chapter 4. It includes a revision of the conditioning of a system and the crucial stability of a spectral problem, which open the doors to both the "a priori" analysis of errors and "a posteriori" analysis of errors. </p>
<p>Chapter 5 and 6 have a more technical character. They include, of course, the famous $QR$ and $QZ$ algorithms, and some numerical methods for "large" matrices: the Lanczos method, and the Arnoldi's method among others. Chapter 7 is devoted to Chebyshev's iterative methods, while polymorphic information processing with matrices is treated in Chapter 8. This includes the homotopic deviation, that is, the study of the analyticity of the map<br />
$$<br />
t\mapsto (A(t)-zI)^{-1}<br />
$$<br />
around $t=0$ and $t=\infty$.</p>
<p>This carefully written graduate-level book constitutes a useful reference for students and researchers in the field. It is a french book written in English language. Consequently, much of it is written in the definition-theorem-proof format, with its emphasis on the best currently available methods for a range of important problems. The bibliography is complemented by bibliographic comments at the end of each chapter. The author is a very recognized expert in the field, and so the selection of the included topics is very accurate.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Jose F. Fernando</div></div></div><div class="field field-name-field-review-legacy-affiliation field-type-text field-label-inline clearfix"><div class="field-label">Affiliation: </div><div class="field-items"><div class="field-item even">Universidad Complutense de Madrid</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book was originally published in two separate volumes by Mason, Paris "Valeurs propres de matrices" (1988) and "Exercises de valeurs propres de matrices" (1989). This SIAM edition is a revised republication of the first english version of the book, published by John Wiley & Sons, Inc., in 1993. It is a work on numerical analysis in depth whose mail goal is to give a modern and complete theory, on an elementary level, of the problem of calculation of eigenvalues of square matrices.</p>
<p>It is carefully written graduate-level book constitutes a useful reference for students and researchers in the field. It is a french book written in English language. Consequently, much of it is written in the definition-theorem-proof format, with its emphasis on the best currently available methods for a range of important problems. The bibliography is complemented by bibliographic comments at the end of each chapter. The author is a very recognized expert in the field, and so the selection of the included topics is very accurate.</p>
<p>Along the book the author uses the language of functional analysis, which has the effect of demonstrating the profound similarity between the different methods of approximation, while the systematic use of bases to represent invariant subspaces provides a geometric interpretation that enhances the traditional algebraic presentation of many algorithms in numerical matrix analysis.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/fran%C3%A7oise-chatelin-exercises-mario-ahu%C3%A9s" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">françoise chatelin. with exercises by mario ahués</a></li><li class="vocabulary-links field-item odd"><a href="/author/fran%C3%A7%C2%8Doise-chatelin-translated-additional-material-walter-ledermannf" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">françoise chatelin. translated with additional material by walter ledermannf</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/philadelphia-siam-society-industrial-and-applied-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">philadelphia: siam, society for industrial and applied mathematics</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2012</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1-611972-45-0</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/15-linear-and-multilinear-algebra-matrix-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">15 Linear and multilinear algebra, matrix theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/15a18" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">15a18</a></li></ul></span>Fri, 07 Feb 2014 09:44:48 +0000Anonymous45552 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/eigenvalues-matrices-fran%C3%A7oise-chatelin-exercises-mario-ahu%C3%A9s-and-fran%C3%A7oise-chatelin#commentsPositive Definite Matrices
https://euro-math-soc.eu/review/positive-definite-matrices
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book represents the first synthesis of the considerable body of new research in positive definite matrices. Through detailed explanations and an authoritative and inspiring writing style, R. Bhatia carefully develops general techniques that have wide applications in the study of such matrices. The book begins with a quick review of some of the basic properties of positive matrices. The author introduces several key topics in functional analysis, operator theory, harmonic analysis and differential geometry, all built around the central theme of positive definite matrices. Chapters 2 and 3 are devoted to a study of positive and completely positive maps and, in particular, on their use in proving inequalities. In chapter 4 the author discusses means of two positive definite matrices with special emphasis on the geometric mean. Among some spectacular applications of these ideas, the author includes proofs of some theorems in the field of matrix convex functions and two of the most famous theorems in the field of quantum mechanical entropy. Chapter 5 gives a quick introduction to positive definite functions on the real line. Again, special attention is given to various means of matrices. Many of these results come from recent research work. Chapter 6 presents some standard and important theorems of Riemannian geometry as seen from the perspective of matrix analysis. Notes and references are appended to each chapter.<br />
The textbook is an informative and useful reference book for mathematicians and other researchers and practitioners. The numerous exercises and notes also make it ideal for graduate-level courses.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">kn</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/r-bhatia" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">r. bhatia</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/princeton-university-press-princeton-princeton-series-applied-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press, princeton: princeton series in applied mathematics</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2006</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-691-12918-1 </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 55 </div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/15-linear-and-multilinear-algebra-matrix-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">15 Linear and multilinear algebra, matrix theory</a></li></ul></span>Sun, 23 Oct 2011 14:37:08 +0000Anonymous40061 at https://euro-math-soc.euPractical Linear Algebra - A Geometry Toolbox
https://euro-math-soc.eu/review/practical-linear-algebra-geometry-toolbox
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book covers all the fundamental topics of linear algebra and outlines some of its applications. Unlike the majority of mathematical textbooks, it does not use the standard theorem-proof approach. The authors explain linear algebra by means of examples and its practical and geometric applications. The reader is led to an intuitive understanding of the notions of the subject and provided with a nice survey of their applications. The basic topics (vectors, linear maps, linear systems, affine maps, and eigenvalues and eigenvectors) are explained first in two dimensions. Then the same concepts are retained and extended in a three dimensional setting, and finally general linear systems of equations and general vector spaces are defined and studied. In addition, several chapters are devoted to geometric and practical applications including the analysis of conics, polygons, triangulations, numerical methods, and curves. The book also contains a brief postscript tutorial chapter and solutions to selected problems. The book is designed for students of fields using linear algebra, such as engineering or computer science. The text could help specialists in these branches to understand the mathematical background of the methods they use. It is also an inspiring book for pure mathematicians who would like to learn more about applications of linear algebra. It is definitely a useful source of many nice examples and applications for teachers of linear algebra.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">pru</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/g-farin" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">g. farin</a></li><li class="vocabulary-links field-item odd"><a href="/author/d-hansford" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">d. hansford</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/k-peters-wellesley-0" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">a. k. peters, wellesley</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2005</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">1-56881-234-5</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 67</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/15-linear-and-multilinear-algebra-matrix-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">15 Linear and multilinear algebra, matrix theory</a></li></ul></span>Sat, 22 Oct 2011 18:09:33 +0000Anonymous39990 at https://euro-math-soc.euIndefinite Linear Algebra and Applications
https://euro-math-soc.eu/review/indefinite-linear-algebra-and-applications
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book is dedicated to relatively recent results in linear algebra of spaces with indefinite inner product. Within this framework, it presents the theory of subspaces and orthogonalization and then goes on to the theory of matrices, perturbation and stability theory. The book also includes applications of the theory to a study of matrix polynomials with selfadjoint constant coefficients, to differential and difference equations with constant coefficients, and to algebraic Ricatti equations.<br />
After the notation and conventions, the book starts with basic geometric ideas concerning spaces with an indefinite inner product, the main topics here being orthogonalization, classification of subspaces and orthogonal polynomials. Further sections are devoted to a study of the classification of linear transformations in indefinite inner product spaces. H-selfadjoint, H-unitary and H-normal transformations together with their canonical forms are of particular interest. Functional calculus is discussed in the next chapter, where special attention is paid to the logarithmic and exponential functions. One chapter is used for a detailed analysis of the structure of H-normal matrices in spaces with an indefinite inner product. Following this, perturbation and stability theories for H-selfadjoint and H-unitary matrices are studied. This topic is important in applications involving the stable boundedness of solutions of differential and difference equations. One section is devoted to applications involving differential equations of the first order, the other for equations of higher orders. The last chapter contains the theory of algebraic Ricatti equations. The appendix serves as a refresher of some parts of linear algebra and matrix theory used in the main body of the book.<br />
Each chapter ends with a series of examples that illustrates the discussed topics. The book has the structure of a graduate text in which chapters on advanced linear algebra form the core. This, together with many significant applications and an accessible style, makes it useful for engineers, scientists and mathematicians alike.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">jsp</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/i-gohberg" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">i. gohberg</a></li><li class="vocabulary-links field-item odd"><a href="/author/p-lancaster" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">p. lancaster</a></li><li class="vocabulary-links field-item even"><a href="/author/l-rodman" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">l. rodman</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/birkh%C3%A4user-basel-0" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">birkhäuser, basel</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2005</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">3-7643-7349-0</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">EUR 38</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/15-linear-and-multilinear-algebra-matrix-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">15 Linear and multilinear algebra, matrix theory</a></li></ul></span>Fri, 21 Oct 2011 18:06:32 +0000Anonymous39948 at https://euro-math-soc.euMatrix Groups for Undergraduates
https://euro-math-soc.eu/review/matrix-groups-undergraduates
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book is written as a textbook for undergraduate students familiar with linear algebra and abstract algebraic structures. It could be used as an excellent textbook for a one semester course at university and it will prepare students for a graduate course on Lie groups, Lie algebras, etc. The author begins with basic facts on matrices (definition, operations and matrices as linear transformations), quaternions, general linear groups and change of basis. In the eight following chapters he explains matrix groups, orthogonal groups, topology of matrix groups, Lie algebras, matrix exponentiation, matrix groups as manifolds, Lie brackets and maximal tori. The book combines an intuitive style of writing (with many examples and a geometric motivation) with rigorous definitions and proofs, giving examples from fields of mathematics, physics and other sciences, where matrices are successfully applied. The book will surely be interesting and helpful for students of algebra and their teachers.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">mbec</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/k-tapp" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">k. tapp</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/american-mathematical-society-providence-student-mathematical-library-vol-29" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">american mathematical society, providence: student mathematical library, vol. 29</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2005</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">0-8218-3785-0</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 29</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/15-linear-and-multilinear-algebra-matrix-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">15 Linear and multilinear algebra, matrix theory</a></li></ul></span>Fri, 21 Oct 2011 12:56:13 +0000Anonymous39932 at https://euro-math-soc.euLinear Algebra in Action
https://euro-math-soc.eu/review/linear-algebra-action
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This is a book on linear algebra, playing an important role in pure and applied mathematics, computer science, physics and engineering. The book is divided into 23 chapters and 2 appendices. The first six chapters and some selected parts from chapters 7-9 are based on classical linear algebra topics. The reader will find here many interesting principles and results concerning vector spaces, Gaussian elimination and its applications, determinants, eigenvalues and eigenvectors, Jordan forms and their calculations, normed linear spaces, inner product spaces and orthogonality, and symmetric, Hermitian and normal matrices. The next three chapters are devoted to singular values and related inequalities, pseudoinverses and triangular factorization and positive definite matrices.<br />
Chapter 13 treats difference equations, differential equations and their systems. Chapters 14-16 contain applications to vector valued functions, the implicit function theorem and extremal problems. The subsequent chapters deal with matrix valued holomorphic functions, matrix equations, realization theory, eigenvalue location problems, zero location problems, convexity and matrices with nonnegative entries. Two appendices describe useful facts from complex function theory. The book offers basic and advanced techniques of linear algebra from the point of view of analysis. Each technique is illustrated by a wide sample of applications and it is accompanied by many exercises of varying difficulty, which give further extensions of the theory. The book can be recommended as a general text for a variety of courses on linear algebra and its applications, as well as a self-study aid for graduate and undergraduate students.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">mbec</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/h-dym" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">h. dym</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/american-mathematical-society-providence-graduate-studies-mathematics-vol-78" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">american mathematical society, providence: graduate studies in mathematics, vol. 78</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2007</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-8218-3813-6 </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 79</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/15-linear-and-multilinear-algebra-matrix-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">15 Linear and multilinear algebra, matrix theory</a></li></ul></span>Fri, 30 Sep 2011 17:30:35 +0000Anonymous39820 at https://euro-math-soc.euMatrix Theory. From Generalized Inverses to Jordan Form
https://euro-math-soc.eu/review/matrix-theory-generalized-inverses-jordan-form
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This book is designed for a “second” course in linear algebra and matrix theory taught at the senior undergraduate and early postgraduate level. It presupposes that the reader has already taken a one-semester course on the elements of linear algebra. The necessary prerequisites are summarized in four appendices. The text is divided into twelve chapters. Chapter 1 is on solutions of systems of linear equations with an emphasis on invertible matrices, and it contains a treatment of the Henderson-Searle formula for the inverse of a sum of matrices and its generalizations. Chapter 2 introduces LU factorization and the Frame algorithm for computing the coefficients of the characteristic polynomial leading to the Cayley-Hamilton theorem. Chapter 3 is on Sylvester’s rank formula and its many consequences. The chapter culminates with the characterization of nilpotent matrices. Left and right inverses are also introduced. </p>
<p>Chapter 4 introduces the main theme of the book: the Moore-Penrose inverse. It is followed in chapter 5 by generalized inverses. A short chapter 6 is about norms followed by chapter 7 on inner products, in particular on the QR factorization and algorithms to find it. The minimum norm and the least square solutions and its connection to the Moore-Penrose inverse are also presented. Chapter 8 discusses orthogonal projections and a connection between the Moore-Penrose inverse and the orthogonal projections on the fundamental subspaces of a matrix. Chapter 9 covers the spectral theorem and chapter 10 covers the primary decomposition theorem, Schur’s triangularization theorem and singular value decomposition. The book then culminates with the Jordan canonical form theorem in chapter 11 and a brief introduction to multilinear algebra in chapter 12. The book contains numerous exercises and homework problems as well as suggestions for further reading. Many concepts are demonstrated with the help of MATLAB.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">jtu</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/r-piziak" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">r. piziak</a></li><li class="vocabulary-links field-item odd"><a href="/author/pl-odell" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">p.l. odell</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/chapman-hallcrc-boca-raton-pure-and-applied-mathematics-vol-288" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">chapman & hall/crc, boca raton: pure and applied mathematics, vol. 288</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2007</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1-58488-625-9 </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 89.95</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/15-linear-and-multilinear-algebra-matrix-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">15 Linear and multilinear algebra, matrix theory</a></li></ul></span>Fri, 30 Sep 2011 16:56:51 +0000Anonymous39798 at https://euro-math-soc.euA (Terse) Introduction to Linear Algebra
https://euro-math-soc.eu/review/terse-introduction-linear-algebra
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>This small booklet published in the series Student Mathematical Library of the AMS is devoted to a basic course on linear algebra as taught in standard undergraduate courses on the topic. It covers (in an economical way) all the necessary topics to be expected (vector spaces, basis and dimension, systems of linear equations, linear maps and their relations to matrices, determinants, linear functionals and adjoint maps, inner-product spaces with a discussion of normal, unitary and orthogonal operators, nilpotent operators and the Jordan canonical form, and characteristic and minimal polynomials of a map). The last chapter also covers the basics of quadratic forms, the Perron-Frobenius theory, stochastic matrices and representations of finite groups in a reduced form. The appendix contains a review of the necessary prerequisites (in particular concerning polynomials). The book is written in an elegant, condensed way. It contains many exercises, mostly of theoretical character. The main advantage (in particular for teachers and talented students) is that basic ideas are carefully isolated and presented in a simple, minimal and understandable way. It is a very good complement to many other books containing calculus, specific examples, and applications of linear algebra.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">vs</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/y-katznelson" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">y. katznelson</a></li><li class="vocabulary-links field-item odd"><a href="/author/yr-katznelson" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">y.r. katznelson</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/american-mathematical-society-providence-student-mathematical-library-vol-44" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">american mathematical society, providence: student mathematical library, vol. 44</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2008</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-8218-4419-9</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 35</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/15-linear-and-multilinear-algebra-matrix-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">15 Linear and multilinear algebra, matrix theory</a></li></ul></span>Fri, 30 Sep 2011 15:48:38 +0000Anonymous39790 at https://euro-math-soc.eu