Showing papers by "Roger A. Horn published in 2009"
TL;DR: In this paper, the authors use methods of the general theory of congruence and *congruence for complex matrices (regularization and cosquares) to determine a unitary *congruence canonical form.
Abstract: We use methods of the general theory of congruence and *congruence for complex matrices – regularization and cosquares – to determine a unitary congruence canonical form (respectively, a unitary *congruence canonical form) for complex matrices A such that ĀA (respectively, A 2) is normal. As special cases of our canonical forms, we obtain – in a coherent and systematic way – known canonical forms for conjugate normal, congruence normal, coninvolutory, involutory, projection, λ-projection, and unitary matrices. But we also obtain canonical forms for matrices whose squares are Hermitian or normal, and other cases that do not seem to have been investigated previously. We show that the classification problems under (a) unitary *congruence when A 3 is normal, and (b) unitary congruence when AĀA is normal, are both unitarily wild, so these classification problems are hopeless.
19 citations
TL;DR: In this article, the authors consider the class of normal complex matrices that commute with their complex conjugate and show that such matrices are real orthogonally similar to a canonical direct sum of 1-by-1 and certain 2-by2 matrices.
9 citations
TL;DR: In this article, Đocovic and Szechtman showed that a vector space V endowed with a bilinear form has determinant 1 if and only if V has no orthogonal summands of odd dimension.
7 citations
TL;DR: In this paper, it was shown that every nonsingular matrix has a ψ S polar decomposition, where R is a symmetric matrix and E is an antiorthogonal matrix.
2 citations
TL;DR: In this paper, the authors give a representation for the factors of the polar decomposition of a nonsingular real square matrix of order 2, and generalized Uhlig's formulae are generalized to encompass all nonzero complex matrices with rank at least n − 1.
Abstract: In [F. Uhlig, Explicit polar decomposition and a near-characteristic polynomial: The 2 × 2 case, Linear Algebra Appl., 38:239–249, 1981], the author gives a representation for the factors of the polar decomposition of a nonsingular real square matrix of order 2. Uhlig’s formulae are generalized to encompass all nonzero complex matrices of order 2 as well as all order n complex matrices with rank at least n − 1.
1 citations
01 Jan 2009
TL;DR: Uhlig's formulae are generalized to encompass all nonzero complex matrices of order 2 as well as all order n complex matrix with rank at least n � 1.
Abstract: In (F. Uhlig, Explicit polar decomposition and a near-characteristic polynomial: The 2 × 2c ase,Linear Algebra Appl., 38:239-249, 1981), the author gives a representation for the factors of the polar decomposition of a nonsingular real square matrix of order 2. Uhlig's formulae are generalized to encompass all nonzero complex matrices of order 2 as well as all order n complex matrices with rank at least n � 1.
1 citations