The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

"Matrix Mathematics" is a reference work for users of matrices in all branches of engineering, science, and applied mathematics. This book brings together a vast body of results on matrix theory for easy reference and immediate application. Each chapter begins with the development of relevant background theory followed by a large collection of specialized results. Hundreds of identities, inequalities, and matrix facts are stated rigorously and clearly with cross references, citations to the literature, and illuminating remarks. Twelve chapters cover all of the major topics in matrix theory: preliminaries; basic matrix properties; matrix classes and transformations; matrix polynomials and rational transfer functions; matrix decompositions; generalized inverses; Kronecker and Schur algebra; positive-semidefinite matrices; norms; functions of matrices and their derivatives; the matrix exponential and stability theory; and linear systems and control theory. A detailed list of symbols, a summary of notation and conventions, an extensive bibliography with author index, and an extensive index are provided for ease of use. The book will be useful for students at both the undergraduate and graduate levels, as well as for researchers and practitioners in all branches of engineering, science, and applied mathematics.

/pdf/matrix-mathematics-theory-facts-and-formulas-with-8swhnd659u.pdf

Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory

This study provides an update of the polarimetric tools currently being used for optimum information extraction from polarimetric synthetic aperture radar (SAR) images. The basics of polarimetric theory are summarized and discussed in the context of SAR. Calibration of polarimetric SAR, which is an important issue for the extraction of meaningful polarization information, is reviewed. Information extraction using the scattered and received wave parameters and target decomposition theory is considered. In particular, the use of coherent versus incoherent target decomposition is discussed and the practical limitations of these target decompositions are outlined. Speckle filtering and classification of polarimetric SAR images are also thoroughly analyzed, and the important directions for future research are outlined.

A review of polarimetry in the context of synthetic aperture radar: concepts and information extraction

Linear preserver problems is an active research area in matrix and operator theory. These problems involve certain linear operators on spaces of matrices or operators. We give a general introduction to the subject in this article. In the first three sections, we discuss motivation, results, and problems. In the last three sections, we describe some techniques, outline a few proofs, and discuss some exceptional results. 1. EXAMPLES AND TYPICAL PROBLEMS. Let Mm,n be the set of m × n complex matrices, and let Mn = Mn,n. Suppose that M, N ∈ Mn satisfy det( MN ) = 1. Then the mapping φ : Mn → Mn given by

/pdf/linear-preserver-problems-5btehedgv2.pdf

Linear Preserver Problems

Theory of rings

A canonical form for matrices under consimilarity

A lower bound for the smallest singular value

Two square complex matrices A B are said to be unitarily congruent if there is a unitary matrix U of the same size such that A=UBUT . We show that A and B are unitarily congruent if and only if there is a unitary V such that . As an application of this characterization, we give simple derivations of several canonical forms for special classes of matrices under unitary congruence.

Yoopyo Hong

Papers

A canonical form for matrices under consimilarity

A lower bound for the smallest singular value

A characterization of unitary congruence

The Jordan cononical form of a product of a Hermitian and a positive semidefinite matrix

A Hermitian canonical form for complex matrices under consimilarity