scholarly journals A note on two-sided removal and cancellation properties associated with Hermitian matrix

2021 ◽  
Author(s):  
Yongge Tian
Keyword(s):  
Hermitian Matrix ◽  
Matrix Product ◽  
Conjugate Transpose ◽  
Square Matrix

Abstract A complex square matrix A is said to be Hermitian if A = A∗, the conjugate transpose of A. We prove that each of the two triple matrix product equalities AA∗A = A∗AA∗ and A3 = AA∗A implies that A is Hermitian by means of decompositions and determinants of matrices, which are named as two-sided removal and cancellation laws associated with a Hermitian matrix. We also present several general removal and cancellation laws as the extensions of the preceding two facts about Hermitian matrix.AMS classifications: 15A24, 15B57

scholarly journals A Schur-Cohn theorem for matrix polynomials

1990 ◽  
Vol 33 (3) ◽  
pp. 337-366 ◽  
Author(s):  
Harry Dym ◽  
Nicholas Young
Keyword(s):  
Unit Circle ◽  
Matrix Polynomial ◽  
Hermitian Matrix ◽  
Krein Space ◽  
Gram Matrix ◽  
Natural Basis ◽  
Test Matrix ◽  
Matrix Polynomials ◽  
Rational Vector ◽  
Square Matrix

Let N(λ) be a square matrix polynomial, and suppose det N is a polynomial of degree d. Subject to a certain non-singularity condition we construct a d by d Hermitian matrix whose signature determines the numbers of zeros of N inside and outside the unit circle. The result generalises a well known theorem of Schur and Cohn for scalar polynomials. The Hermitian “test matrix” is obtained as the inverse of the Gram matrix of a natural basis in a certain Krein space of rational vector functions associated with N. More complete results in a somewhat different formulation have been obtained by Lerer and Tismenetsky by other methods.

A trace bound for a general square matrix product

2000 ◽  
Vol 45 (8) ◽  
pp. 1563-1569 ◽  
Author(s):  
Wei Xing ◽  
Qingling Zhang ◽  
Qiyi Wang
Keyword(s):  
Matrix Product ◽  
Square Matrix

A New Trace Bound for a General Square Matrix Product

2007 ◽  
Vol 52 (2) ◽  
pp. 349-352 ◽  
Author(s):  
Jianzhou Liu ◽  
Lingli He
Keyword(s):  
Matrix Product ◽  
Square Matrix

On the Solutions of the Matrix Equation f(X,X*)=g(X,X*)

1972 ◽  
Vol 15 (1) ◽  
pp. 45-49
Author(s):  
P. Basavappa
Keyword(s):  
Matrix Equation ◽  
Identity Matrix ◽  
Normal Matrices ◽  
Special Cases ◽  
The Matrix ◽  
Conjugate Transpose ◽  
Matrix Identities ◽  
Square Matrix

It is well known that the matrix identities XX*=I, X=X* and XX* = X*X, where X is a square matrix with complex elements, X* is the conjugate transpose of X and I is the identity matrix, characterize unitary, hermitian and normal matrices respectively. These identities are special cases of more general equations of the form (a)f(X, X*)=A and (b)f(Z, X*)=g(X, X*) where f(x, y) and g(x, y) are monomials of one of the following four forms: xyxy…xyxy, xyxy…xyx, yxyx…yxyx, and yxyx…yxy. In this paper all equations of the form (a) and (b) are solved completely. It may be noted a particular case of f(X, X*)=A, viz. XX'=A, where X is a real square matrix and X' is the transpose of X was solved by WeitzenbÖck [3]. The distinct equations given by (a) and (b) are enumerated and solved.

A note on some theorems on simultaneous diagonalization of two Hermitian matrices

1971 ◽  
Vol 70 (3) ◽  
pp. 383-386 ◽  
Author(s):  
Yik-Hoi Au-Yeung
Keyword(s):  
Hermitian Matrix ◽  
Complex Numbers ◽  
Single Element ◽  
Singular Matrix ◽  
Simultaneous Diagonalization ◽  
Hermitian Matrices ◽  
Real Numbers ◽  
Skew Field ◽  
Conjugate Transpose ◽  
Real Quaternions

We denote by F the field R of real numbers, the field C of complex numbers, or the skew-field H of real quaternions, and by Fn an n-dimensional left vector space over F. If A is a matrix with elements in F, we denote by A* its conjugate transpose. In all three cases of F, an n × n matrix A is said to be Hermitian if A = A* and unitary if AA* = In, where In is the n × n identity matrix. An n × n Hermitian matrix A is said to be positive definite (postive semi-definite resp.) if uAu* > 0(uAu* ≥ 0 resp.) for all u (╪ 0) in Fn. Here and in what follows we regard u as a 1 × n matrix and identify a 1 × 1 matrix with its single element. In the following we shall always use A and B to denote two n×n Hermitian matrices with elements in F, and we say that A and B can be diagonalized simultaneously if there exists an n×n non-singular matrix V with elements in F such that VAV* and VBV* are diagonal matrices. We shall use diag {A1, A2} to denote a diagonal block matrix with the square matrices A1 and A2 lying on its diagonal.

scholarly journals A characterization of the definiteness of a Hermitian matrix

1974 ◽  
Vol 15 (1) ◽  
pp. 1-4
Author(s):  
Yik-Hoi Au-Yeung ◽  
Tai-Kwok Yuen
Keyword(s):  
Hermitian Matrix ◽  
Nonsingular Matrix ◽  
Complex Numbers ◽  
Identity Matrix ◽  
Hermitian Matrices ◽  
Real Numbers ◽  
Skew Field ◽  
Conjugate Transpose ◽  
Real Quaternions

We denote by F the field R of real numbers, the field C of complex numbers or the skew-field H of real quaternions, and by Fn an n-dimensional left vector space over F. If A is a matrix with elements in F, we denote by A* its conjugate transpose. In all three cases of F, an n × n matrix A is said to be hermitian (unitary resp.) if A = A* (AA*= identity matrix resp.). An n ×x n hermitian matrix A is said to be definite (semidefinite resp.) if uAu*vAv* ≥ 0 (uAu*vAv* ≧ 0 resp.) for all nonzero u and v in Fn. If A and B are n × n hermitian matrices, then we say that A and B can be diagonalized simultaneously into blocks of size less than or equal to m (abbreviated to d. s. ≧ m) if there exists a nonsingular matrix U with elements in F such that UAU* = diag{A1,…, Ak} and UBU* = diag{B1…, Bk}, where, for each i = 1, …, k, Ai and Bk are of the same size and the size is ≧ m. In particular, if m = 1, then we say A and B can be diagonalized simultaneously (abbreviated to d. s.).

Stochastic local operations and classical communication invariants via square matrix

Laser Physics ◽  
2019 ◽  
Vol 29 (2) ◽  
pp. 025203 ◽  
Author(s):  
Xinwei Zha ◽  
Irfan Ahmed ◽  
Da Zhang ◽  
Wen Feng ◽  
Yanpeng Zhang
Keyword(s):  
Classical Communication ◽  
Square Matrix
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