Re: [math-fun] Matrix Arithmetic Compendium
There is 30-page booklet by Marcus that has a really excellent summary of matrix theory. From MathSciNet: [4] MR0109824 (22 #709) Marcus, Marvin Basic theorems in matrix theory. Nat. Bur. Standards Appl. Math. Ser. 57 1960 iv+27 pp. (Reviewer: A. S. Householder) 15.00 For anyone with even a passing interest in the subject, this is certainly the best book bargain one could imagine! It is a compact collection of definitions, identities, inequalities, and basic theorems, grouped into six chapters, as follows: General definitions and elementary properties; Canonical forms, invariance, congruence, commutativity, orthogonalization; Eigenvalues, determinants, submatrices, rank; Determinant and rank inequalities; Linear equations $Ax=b$; Inversion of $A$; Some particular matrices and their condition numbers. Each of these is well subdivided, so that there is no great difficulty in finding what is there even though the index is perhaps less complete than one might like. While the collection is to be recommended enthusiastically, it is, nevertheless, not beyond criticism. Statements are often referenced, but by no means always. The list of 47 references is quite meager, and the selection seems a bit haphazard. Localization theorems of the Perron-Gersch-gorin-Brauer type are given, but there is no mention of the Weinstein theorem. The Givens method for computing eigenvalues of a hermitian matrix is stated after Jacobi's method, and is called "the Givens' modification", whereas the two methods have only superficial resemblance. The condition for convergence of the Jacobi iteration for solving a system of equations is incorrectly stated (top of p. 18). It is to be hoped, however, that the author will consider subsequent expansions, with appropriate revision. Reviewed by A. S. Householder NJAS
Also: M. Marcus and H. Minc, A Survey of Matrix Theory and Matrix Inequalities, Dover, NY, 1992 (originally published in the sixties). E. Deutsch
Also: R.T. Gregory and D.L. Karney, A collection of matrices for testing computational algorithms, Robert E. Krieger Publishing Company, Huntington, New York, 1978. R. T. Gregory and D. L. Karney. A Collection of Matrices for Testing Computational Algorithms. John Wiley and Sons, New York, 1969. I also recall vaguely a collection of matrices by A. S. Householder in a (blue?) paperback publication, possibly by Argonne Nat. Lab. ??? E. Deutsch
The site http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/index1.html offers (free) summaries and definitions of hundreds of matrix types. This is not what Rich asked for, but it might be useful anyway and is good to know about. Steve Gray ----- Original Message ----- From: "N. J. A. Sloane" <njas@research.att.com> To: <math-fun@mailman.xmission.com>; "math-fun" <math-fun@mailman.xmission.com> Cc: <rcs@cs.arizona.edu>; <njas@research.att.com> Sent: Wednesday, October 29, 2003 5:34 AM Subject: Re: [math-fun] Matrix Arithmetic Compendium
There is 30-page booklet by Marcus that has a really excellent summary of matrix theory. From MathSciNet:
[4] MR0109824 (22 #709) Marcus, Marvin Basic theorems in matrix theory. Nat. Bur. Standards Appl. Math. Ser. 57 1960 iv+27 pp. (Reviewer: A. S. Householder) 15.00
For anyone with even a passing interest in the subject, this is certainly the best book bargain one could imagine! It is a compact collection of definitions, identities, inequalities, and basic theorems, grouped into six chapters, as follows: General definitions and elementary properties; Canonical forms, invariance, congruence, commutativity, orthogonalization; Eigenvalues, determinants, submatrices, rank; Determinant and rank inequalities; Linear equations $Ax=b$; Inversion of $A$; Some particular matrices and their condition numbers. Each of these is well subdivided, so that there is no great difficulty in finding what is there even though the index is perhaps less complete than one might like. While the collection is to be recommended enthusiastically, it is, nevertheless, not beyond criticism. Statements are often referenced, but by no means always. The list of 47 references is quite meager, and the selection seems a bit haphazard. Localization theorems of the Perron-Gersch-gorin-Brauer type are given, but there is no mention of the Weinstein theorem. The Givens method for computing eigenvalues of a hermitian matrix is stated after Jacobi's method, and is called "the Givens' modification", whereas the two methods have only superficial resemblance. The condition for convergence of the Jacobi iteration for solving a system of equations is incorrectly stated (top of p. 18). It is to be hoped, however, that the author will consider subsequent expansions, with appropriate revision. Reviewed by A. S. Householder
NJAS
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participants (3)
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Emeric Deutsch -
N. J. A. Sloane -
Steve Gray