A canonical form for nonderogatory matrices under unitary similarity

Similarity transformation of matrices to one common canonical form and its applications to 2D linear systems

The notion of a common canonical form for a sequence of square matrices is introduced. Necessary and sufficient conditions for the existence of a similarity transformation reducing the sequence of matrices to the common canonical form are established. It is shown that (i) using a suitable state vector linear transformation it is possible to decompose a linear 2D system into two linear 2D subsys...

Crossing Matrices and Thurston's Canonical Form for Braids

Eigenvalues of the Sum of Matrices from Unitary Similarity Orbits

Abstract Let A and B be n×n complex matrices. Characterization is given for the set E(A,B) of eigenvalues of matrices of the form UAU + V BV for some unitary matrices U and V . Consequences of the results are discussed and computer algorithms and programs are designed to generate the set E(A,B). The results refine those of Wielandt on normal matrices. Extensions of the results to the sum of mat...

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...