A canonical form for H-unitary matrices

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...

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 )], ...

Crossing Matrices and Thurston's Canonical Form for Braids

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...

Canonical Forms for Unitary Congruence and *Congruence

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) is normal. As special cases of our canonical forms, we obtain—in a coherent and systematic way—known canonical forms for con...