##
**Topics in matrix analysis.**
*(English)*
Zbl 0729.15001

Cambridge etc.: Cambridge University Press. viii, 607 p. £45.00; $ 59.50 (1991).

This book is a sequel to the authors’ Matrix analysis (1985; Zbl 0576.15001) and followed by the Russian translation in 1989. Most of the material is not covered by other books, and the authors are particularly thorough in presenting motivations or applications of the concepts studied as well as in tracing their historical development. One of the really fascinating is the history, perhaps not well known in general, of singular values discovered in several different ways, namely in at least three approaches to orthogonal substitutions of bilinear forms in the period 1873-1890 by E. Beltrami, C. Jordan, J. J. Sylvester, and in the work of E. Schmidt (1907) concerning integral equations. This predestined the prominent role of singular values and their connections with eigenvalues in the subsequent development (I. Schur, É. Picard, F. Smithies, S.-H. Chang, H. Weyl, Ky Fan, A. Horn) culminating in Yamamoto’s formulas (1967) showing the asymptotic exactness of the former estimates. Chapter 3 on 105 pages gives a good picture of this one of the most brilliant achievements of mathematics.

But also the other chapters are written in this style. Before formal introducing the field of values (the numerical range) in chapter 1, four motivations for it are given. Its elegant theory is then presented on 88 pages including the Toeplitz-Hausdorff convexity theorem with three proofs, a Gershgorin-type localization for the field of values, the relationships between the boundary points and eigenvalues, an intersection characterization by normal dilations, connections with the real part of the operator, behaviour with respect to the products of operators, an axiomatic characterization, and various generalizations.

Chapter 2 (45 pages) deals with the notion of stable matrices and gives a criterion in terms of the Lyapunov equation. It also contains a thorough study of an important special class, the so called M-matrices, with a list of eighteen equivalent characterizations.

In chapter 4 (59 pages) the Kronecker product is used to obtain spectral criteria for solvability of linear matrix equations, including the Lyapunov equation. The derivations on \(M_ n({\mathbb{C}})\) are proven to be inner, and the second commutant of a general matrix A is shown to coincide with the polynomials in A. The Shoda characterizations (1936) of the matrix commutators, both additive and multiplicative, are given as well as the Frobenius characterizations (1897) of the linear preservers on \(M_ n({\mathbb{C}})\) with respect to the determinant or spectrum.

The Hadamard-Schur (entrywise) product arises in many natural ways, for instance, if one wants to express the diagonal of a diagonalizable matrix in terms of the eigenvalues. In some problems this product becomes more adequate than the usual (operator) product but, surprisingly, it turns out that both products are closely related. A remarkable theorem due to the authors (1987) shows that a unitarily invariant norm on \(M_ n({\mathbb{C}})\) is (or is not) submultiplicative with respect to both products simultaneously. Also the singular values fit in well with the entrywise product. The class of positive definite matrices is closed under Hadamard-Schur multiplication but not under ordinary matrix multiplication. The behaviour of other classes (nonnegative matrices, M- matrices) with respect to Hadamard-Schur multiplication is also studied. Altogether chapter 5 occupies 84 pages.

The last chapter 6 (179 pages) is devoted to various ways in which matrices and functions interact. For instance, an elegant criterion in terms of the ranks of the iterates of A is given in order that A have a square root. Many formulas involving the trace and/or the exponentials occur, and even “nonlinear traces” are considered.

The preceding remarks indicate just a few of the major topics around which the book concentrates. A great deal of very interesting material, including alternate proofs of main theorems, can be found in over 650 problems some of which are equipped with hints. Many results come from current research in which the authors are taking active part. Even interesting open problems are occasionally mentioned, and the references with brief comments can serve as a guide for the literature that could not be included.

To sum up: the present book together with its predecessor approach a modern encyclopedia of matrix analysis. Both are indispensable for the researchers and at the same time available for the beginners.

But also the other chapters are written in this style. Before formal introducing the field of values (the numerical range) in chapter 1, four motivations for it are given. Its elegant theory is then presented on 88 pages including the Toeplitz-Hausdorff convexity theorem with three proofs, a Gershgorin-type localization for the field of values, the relationships between the boundary points and eigenvalues, an intersection characterization by normal dilations, connections with the real part of the operator, behaviour with respect to the products of operators, an axiomatic characterization, and various generalizations.

Chapter 2 (45 pages) deals with the notion of stable matrices and gives a criterion in terms of the Lyapunov equation. It also contains a thorough study of an important special class, the so called M-matrices, with a list of eighteen equivalent characterizations.

In chapter 4 (59 pages) the Kronecker product is used to obtain spectral criteria for solvability of linear matrix equations, including the Lyapunov equation. The derivations on \(M_ n({\mathbb{C}})\) are proven to be inner, and the second commutant of a general matrix A is shown to coincide with the polynomials in A. The Shoda characterizations (1936) of the matrix commutators, both additive and multiplicative, are given as well as the Frobenius characterizations (1897) of the linear preservers on \(M_ n({\mathbb{C}})\) with respect to the determinant or spectrum.

The Hadamard-Schur (entrywise) product arises in many natural ways, for instance, if one wants to express the diagonal of a diagonalizable matrix in terms of the eigenvalues. In some problems this product becomes more adequate than the usual (operator) product but, surprisingly, it turns out that both products are closely related. A remarkable theorem due to the authors (1987) shows that a unitarily invariant norm on \(M_ n({\mathbb{C}})\) is (or is not) submultiplicative with respect to both products simultaneously. Also the singular values fit in well with the entrywise product. The class of positive definite matrices is closed under Hadamard-Schur multiplication but not under ordinary matrix multiplication. The behaviour of other classes (nonnegative matrices, M- matrices) with respect to Hadamard-Schur multiplication is also studied. Altogether chapter 5 occupies 84 pages.

The last chapter 6 (179 pages) is devoted to various ways in which matrices and functions interact. For instance, an elegant criterion in terms of the ranks of the iterates of A is given in order that A have a square root. Many formulas involving the trace and/or the exponentials occur, and even “nonlinear traces” are considered.

The preceding remarks indicate just a few of the major topics around which the book concentrates. A great deal of very interesting material, including alternate proofs of main theorems, can be found in over 650 problems some of which are equipped with hints. Many results come from current research in which the authors are taking active part. Even interesting open problems are occasionally mentioned, and the references with brief comments can serve as a guide for the literature that could not be included.

To sum up: the present book together with its predecessor approach a modern encyclopedia of matrix analysis. Both are indispensable for the researchers and at the same time available for the beginners.

Reviewer: J.Zemánek (Warszawa)

### MathOverflow Questions:

A Handbook of Matrix FactorizationsBounding the norm of a contraction matrix

### MSC:

15-02 | Research exposition (monographs, survey articles) pertaining to linear algebra |

15A18 | Eigenvalues, singular values, and eigenvectors |

15A24 | Matrix equations and identities |

15A27 | Commutativity of matrices |

15A42 | Inequalities involving eigenvalues and eigenvectors |

15A45 | Miscellaneous inequalities involving matrices |

15B48 | Positive matrices and their generalizations; cones of matrices |

15A60 | Norms of matrices, numerical range, applications of functional analysis to matrix theory |

15A69 | Multilinear algebra, tensor calculus |

53A45 | Differential geometric aspects in vector and tensor analysis |