Abstract
In this paper, new notions of the weighted core-EP left inverse and the weighted MPD inverse which are dual to the weighted core-EP (right) inverse and the weighted DMP inverse, respectively, are introduced and represented. The direct methods of computing the weighted right and left core-EP, DMP, MPD, and CMP inverses by obtaining their determinantal representations are given. A numerical example to illustrate the main result is given.
Highlights
In the whole article, R and C stand for fields of the real and complex numbers, respectively
The W-Weighted DMP and MPD Inverses and Their Determinantal Representations e concept of the DMP inverse was introduced by Malik and ome as follows
Let A ∈ Cm×n, W ∈ Cn×m be a nonzero matrix, and k max{Ind(WA), Ind(AW)}. en, the matrix X A†AWAd,WW is the unique solution to the equations: AX AWAd,WW, (46)
Summary
R and C stand for fields of the real and complex numbers, respectively. The direct methods of computing the weighted right and left core-EP, DMP, MPD, and CMP inverses by obtaining their determinantal representations are given. En, the right weighted core-EP inverse A○†,W,r (ai○†j,W,r) ∈ Cm×n possesses the determinantal representations: a○i†j,W,r. I. theorem on the determinantal representation of the left W-weighted core-EP inverse can be proved. Theorem on the determinantal representation of the left W-weighted core-EP inverse can be proved A matrix X ∈ Cn×n is said to be the MPD inverse of A if it satisfies the conditions: AX AAd, 4. The W-Weighted DMP and MPD Inverses and Their Determinantal Representations e concept of the DMP inverse was introduced by Malik and ome as follows. (3) Compute the matrix Ω ≔ ΩWAA∗. (4) find adij,†,W by (50) for all i 1, . . . , m and j 1, . . . , n
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