An alternative canonical form for quaternionic H-unitary matrices

Linear maps transforming H-Unitary Matrices

Let H1 be an n × n invertible Hermitian matrix, and let U(H1) be the group of n × n H1-unitary matrices, i.e., matrices A satisfying A H1A = H1. Suppose H2 is an m × m invertible Hermitian matrix. We show that a linear transformation φ : Mn → Mm satisfies φ(U(H1)) ⊆ U(H2) if and only if there exist invertible matrices S ∈ Mm, U, V ∈ U(H2) such that SH2S = [(Ia ⊕−Ib)⊗H1]⊕ [(Ic ⊕−Id)⊗ (H−1 1 )], ...

H-Unitary and Lorentz Matrices: A Review

Many properties of H-unitary and Lorentz matrices are derived using elementary methods. Complex matrices which are unitary with respect to the indefinite inner product induced by an invertible Hermitian matrix H, are called H-unitary, and real matrices that are orthogonal with respect to the indefinite inner product induced by an invertible real symmetric matrix, are called Lorentz. The focus i...

Crossing Matrices and Thurston's Canonical Form for Braids

Localization theorems for eigenvalues of quaternionic matrices

Ostrowski type and Brauer type theorems are derived for the left eigenvalues of quaternionic matrix. We see that the above theorems for the left eigenvalues are also true for the case of right eigenvalues, when the diagonals of quaternionic matrix are real. Some distribution theorems are given in terms of ovals of Cassini that are sharper than the Ostrowski type theorems, respectively, for the ...

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...