Wyniki wyszukiwania

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Consimilarity and quaternion matrix equations AX −^X B = C, X − A^X B = C

100%

Klimchuk T., Sergeichuk V. V.

Special Matrices

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2014

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tom 2

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nr 1

EN

L. Huang [Consimilarity of quaternion matrices and complex matrices, Linear Algebra Appl. 331 (2001) 21–30] gave a canonical form of a quaternion matrix with respect to consimilarity transformations A ↦ ˜S−1AS in which S is a nonsingular quaternion matrix and h = a + bi + cj + dk ↦ ˜h := a − bi + cj − dk (a, b, c, d ∈ ℝ). We give an analogous canonical form of a quaternion matrix with respect to consimilarity transformations A ↦^S−1AS in which h ↦ ^h is an arbitrary involutive automorphism of the skew field of quaternions. We apply the obtained canonical form to the quaternion matrix equations AX −^X B = C and X − A^X B = C.

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On the Yang-Baxter-like matrix equation for rank-two matrices

100%

Zhou D., Chen G., Ding J.

Open Mathematics

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2017

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tom 15

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nr 1

340-353

EN

Let A = PQT, where P and Q are two n × 2 complex matrices of full column rank such that QTP is singular. We solve the quadratic matrix equation AXA = XAX. Together with a previous paper devoted to the case that QTP is nonsingular, we have completely solved the matrix equation with any given matrix A of rank-two.

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Inertias and ranks of some Hermitian matrix functions with applications

68%

Zhang X., Wang Q.-W., Liu X.

Open Mathematics

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2012

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tom 10

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nr 1

329-351

EN

Let S be a given set consisting of some Hermitian matrices with the same size. We say that a matrix A ∈ S is maximal if A − W is positive semidefinite for every matrix W ∈ S. In this paper, we consider the maximal and minimal inertias and ranks of the Hermitian matrix function f(X,Y) = P − QXQ* − TYT*, where * means the conjugate and transpose of a matrix, P = P*, Q, T are known matrices and for X and Y Hermitian solutions to the consistent matrix equations AX =B and YC = D respectively. As applications, we derive the necessary and sufficient conditions for the existence of maximal matrices of $$H = \{ f(X,Y) = P - QXQ* - TYT* : AX = B,YC = D,X = X*, Y = Y*\} .$$ The corresponding expressions of the maximal matrices of H are presented when the existence conditions are met. In this case, we further prove the matrix function f(X,Y)is invariant under changing the pair (X,Y). Moreover, we establish necessary and sufficient conditions for the system of matrix equations $$AX = B, YC = D, QXQ* + TYT* = P$$ to have a Hermitian solution and the system of matrix equations $$AX = C, BXB* = D$$ to have a bisymmetric solution. The explicit expressions of such solutions to the systems mentioned above are also provided. In addition, we discuss the range of inertias of the matrix functions P ± QXQ* ± TYT* where X and Y are a nonnegative definite pair of solutions to some consistent matrix equations. The findings of this pape extend some known results in the literature.

Ograniczanie wyników

2 Open Mathematics

1 Special Matrices

1 Chen G.

1 Ding J.

1 Klimchuk T.

1 Liu X.

1 Sergeichuk V. V.

1 Wang Q.-W.

1 Zhang X.

1 Zhou D.

1 2017

1 2014

1 2012

Consimilarity and quaternion matrix equations AX −^X B = C, X − A^X B = C

On the Yang-Baxter-like matrix equation for rank-two matrices

Inertias and ranks of some Hermitian matrix functions with applications