Abstract
Based on the core-EP decomposition, we use the WG inverse, Drazin inverse, and other inverses to give some new characterizations of the WG matrix. Furthermore, we generalize the Cayley–Hamilton theorem for special matrices including the WG matrix. Finally, we give examples to verify these results.
Highlights
IntroductionErefore, we call it a group invertible matrix with index 1. e symbol CCn M stands for the set of group invertible matrices in Cn,n: CCn M ArkA2 rk(A), A ∈ Cn,n
Let Cm,n stand for the set of m × n complex matrices. e symbols A∗, R(A), rk(A), and det(A) represent the conjugate transpose, range, rank, and determinant of A, respectively. e smallest positive integer k such that rk(Ak+1) rk(Ak) is called the index of A ∈ Cm,n and it is denoted by Ind(A). e Moore–Penrose inverse of A ∈ Cm,n is the unique matrix X ∈ Cn,m satisfying the following equations: AXA A, XAX X, (AX)∗ AX, (1)
When k 1, X is called the group inverse of A and is denoted by X A#. Erefore, we call it a group invertible matrix with index 1. e symbol CCn M stands for the set of group invertible matrices in Cn,n: CCn M ArkA2 rk(A), A ∈ Cn,n
Summary
Erefore, we call it a group invertible matrix with index 1. e symbol CCn M stands for the set of group invertible matrices in Cn,n: CCn M ArkA2 rk(A), A ∈ Cn,n. Baksalary and Trenkler [3] defined the core inverse of a complex matrix with index 1. Let A ∈ CCn M; the core inverse of A is the unique matrix which satisfies the following equations: AX AA○†, (5). With the development of new generalized inverses, new research tools such as matrix decomposition and algorithm are given. In [26], Wang, Chen, and Yan gave the generalized Cayley–Hamilton theorem of core-EP inverse matrix and DMP inverse matrix by core-EP decomposition, and the characteristic polynomial equations of core-EP inverse matrix and DMP inverse matrix were discussed. Based on the above researches, this paper will focus on the WG matrix, the equivalent characterizations of WG matrix, and the generalized Cayley–Hamilton theorem for special matrices including WG matrix
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