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A new rank formula for idempotent matrices with applications
Yong Ge Tian,George P. H. Styan +1 more
- Vol. 43, Iss: 2, pp 379-384
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TLDR
In this paper, it was shown that the product PA = AA + is the orthogonal projector on the range (column space) of A, where A+ is the Moore-Penrose inverse of A; which is the unique solution of the following four Penrose equations.Abstract:
A complex square matrix A is said to be idempotent, or a projector, whenever A2 = A; when A is idempotent, and Hermitian (or real symmetric), it is often called an orthogonal projector, otherwise an oblique projector. Projectors are closely linked to generalized inverses of matrices. For example, for a given matrix A the product PA = AA + is well known as the orthogonal projector on the range (column space) of A, where A+ is the Moore-Penrose inverse of A; which is the unique solution of the following four Penrose equationsread more
Citations
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Twosetsofnewcharacterizationsfornormaland EP matrices
Shizhen Cheng,Yongge Tian +1 more
TL;DR: In this article, two sets of new characterizations for normal matrices and EP matrices are presented, which are derived through ranks and generalized inverses of matrices, respectively.
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Rank equalities for idempotent matrices with applications
Yongge Tian,George P. H. Styan +1 more
TL;DR: In this paper, elementary block matrix operations are used to derive rank equalities related to sums of two idempotent matrices, and various consequences and applications are given for their applications.
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Some properties of submatrices in a solution to the matrix equation AXB=C with applications
TL;DR: From these formulas, necessary and sufficient conditions for the submatrices to be zero and nonsingular, respectively are derived.
Journal ArticleDOI
Inequalities and equalities for ℓ = 2 (Sylvester), ℓ = 3 (Frobenius), and ℓ > 3 matrices
TL;DR: In this paper, simple proofs of the Sylvester (l = 2) and Frobenius inequalities are given, and a new sufficient condition for the equality of the FFR is provided.
Journal ArticleDOI
Miscellaneous equalities for idempotent matrices with applications
TL;DR: In this article, the authors present the basic mathematical ideas and methodologies of the matrix analytic theory in a readable, up-to-date, and comprehensive manner, including constructions of various algebraic matrix identities composed by the conventional operations of idempotent matrices, and uses of the block matrix method in the derivation of closed-form formulas for calculating the ranks of matrix expressions that are composed by idempotsent matrix.
References
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Journal Article
Some comments on several matrix inequalities with applications to canonical correlations: Historical background and recent developments
TL;DR: In this paper, the authors present a generalized matrix Frucht-Kantorovich inequality and show that it is "essentially equivalent" to the generalized matrix Wielandt inequality given by Lu (1999).