A Hermitian canonical form for complex matrices under consimilarity

Complex-Valued Neural Network for Hermitian Matrices

This paper proposes neural network for computing the eigenvectors of Hermitian matrices. For the eigenvalues of Hermitian matrices, we establish an explicit representation for the solution of the neural network based on Hermitian matrices and analyze its convergence property. We also consider to compute the eigenvectors of skew-symmetric matrices and skew-Hermitian matrices, corresponding to th...

Congruence of Hermitian Matrices by Hermitian Matrices

Two Hermitian matrices A, B ∈ Mn(C) are said to be Hermitian-congruent if there exists a nonsingular Hermitian matrix C ∈ Mn(C) such that B = CAC. In this paper, we give necessary and sufficient conditions for two nonsingular simultaneously unitarily diagonalizable Hermitian matrices A and B to be Hermitian-congruent. Moreover, when A and B are Hermitian-congruent, we describe the possible iner...

Crossing Matrices and Thurston's Canonical Form for Braids

Ela Change of the *congruence Canonical Form of 2-by-2 Matrices under Perturbations

It is constructed the Hasse diagram for the closure ordering on the sets of *congruence classes of 2× 2 matrices. In other words, it is constructed the directed graph whose vertices are 2× 2 canonical complex matrices for *congruence and there is a directed path from A to B if and only if A can be transformed by an arbitrarily small perturbation to a matrix that is *congruent to B.

Unitary Matrices and Hermitian Matrices

Unitary Matrices and Hermitian Matrices Recall that the conjugate of a complex number a + bi is a − bi. The conjugate of a + bi is denoted a+ bi or (a+ bi)∗. In this section, I’ll use ( ) for complex conjugation of numbers of matrices. I want to use ( )∗ to denote an operation on matrices, the conjugate transpose. Thus, 3 + 4i = 3− 4i, 5− 6i = 5 + 6i, 7i = −7i, 10 = 10. Complex conjugation sati...