EP matrix
An EP matrix (or range-Hermitian matrix[1]) is a matrix A whose range is equal to the range of its conjugate transpose A*.
EP matrices were introduced in 1950 by Hans Schwerdtfeger,[1][2] and since then, many equivalent characterizations of EP matrices have been investigated through the literature.[3] The meaning of the EP abbreviation stands originally for Equal Principal, but it is widely believed that it stands for Equal Projectors instead, since an equivalent characterization of EP matrices is based in terms of equality of the projectors AA+ and A+A.[4]
According to the fundamental theorem of linear algebra, the range of any matrix A is equal to the null-space of A*, but is not necessarily equal to the null-space of A. When A is an EP matrix, the range of A is precisely equal to the null-space of A.
Properties
- An equivalent characterization of an EP matrix A is that A commutes with its Moore-Penrose pseudoinverse, that is, the projectors AA+ and A+A are equal. This is similar to the characterization of normal matrices where A commutes with its conjugate transpose.[3]
- The sum of EP matrices Ai is an EP matrix if the null-space of the sum is contained in the null-space of each matrix Ai.[5]
- To be an EP matrix is a necessary condition for normality: A is normal if and only if A is EP matrix and AA*A2 = A2 A*A.[3]
- When A is an EP matrix, the Moore-Penrose pseudoinverse of A is equal to the group inverse of A.[3]
- A is an EP matrix if and only if the Moore-Penrose pseudoinverse of A is an EP matrix.[3]
References
- ^ a b Drivaliaris, Dimosthenis; Karanasios, Sotirios; Pappas, Dimitrios (2008-10-01). "Factorizations of EP operators". Linear Algebra and its Applications. 429 (7): 1555–1567. doi:10.1016/j.laa.2008.04.026. ISSN 0024-3795.
- ^ Schwerdtfeger, Hans (1950). Introduction to linear algebra and the theory of matrices. P. Noordhoff.
- ^ a b c d e Cheng, Shizhen; Tian, Yongge (2003-12-01). "Two sets of new characterizations for normal and EP matrices". Linear Algebra and its Applications. 375: 181–195. doi:10.1016/S0024-3795(03)00650-5. ISSN 0024-3795.
- ^ S., Bernstein, Dennis (2018). Scalar, Vector, and Matrix Mathematics : Theory, Facts, and Formulas. Princeton: Princeton University Press. ISBN 9781400888252. OCLC 1023540775.
{{cite book}}: CS1 maint: multiple names: authors list (link) - ^ Meenakshi, A.R. (1983). "On sums of EP matrices". Houston Journal of Mathematics. 9.