The star partial order and the eigenprojection at 0 on EP matrices

Abstract

The star partial order and the eigenprojection at 0 on EPThe space of n × n complex matrices with the star partial order is considered in the first part of this paper. The class of EP matrices is analyzed and several properties related to this order are given. In addition, some information about predecessors and successors of a given EP matrix is obtained. The second part is dedicated to the study of some properties that relate the eigenprojection at 0 with the star and sharp partial orders.

On a partial order defined by the weighted Moore–Penrose inverse

“…A similar result to Theorem 4.7 in [11] can be stated for EP (M,M ) where the (M, M )-star partial order is used to compare the following pairs of matrices: A and C α,β , C α,β and B, C α,β and C γ,δ , f (C α,β ) and f (A), f (B) and f (C α,β ), where α, β, γ, δ ∈ C. …”

“…Hermitian by Lemma 3.1 and Lemma 4.3 in [11]. Therefore, applying Lemma 3.1 (d) we get that A π is an (M, M )-Hermitian matrix.…”

“…By Remark 4.1 in [11], this last equality holds if and only if Ψ (M,M ) (A) is an orthogonal projector. Now, the proof follows applying Lemma 3.1 (e).…”

“…Then ,M ) (A)). Applying Lemma 4.5 in [11] we get that Ψ (M,M ) (B) is an orthogonal proyector of size n × n given by…”

A simultaneous canonical form of a pair of matrices and applications involving the weighted Moore–Penrose inverse

“…We remind that a pair of matrices A, B ∈ C m×n are ordered under the star order ≤ * , and written A ≤ * B, if AA * = BA * and A * A = A * B [7,8,11,13]. It is well-known that inequalities under ≤ * are preserved under unitary equivalences, that is A ≤ * B if and only if SAT ≤ * SBT for all unitary matrices S ∈ C m×m and T ∈ C n×n .…”