In this paper,the GH-congruence canonical forms of positive semidefinite and definte inite and definite(need not be self-conjugate)quaternion matrices are given,and a neccessary and sufficientcondition of GH-congruenc...In this paper,the GH-congruence canonical forms of positive semidefinite and definte inite and definite(need not be self-conjugate)quaternion matrices are given,and a neccessary and sufficientcondition of GH-congruence for two positive semidifinite(definite)quaternion matrices isgiven also.Then simultaneous GH-congruence reduced forms for two self-conjugate matri-ces and some result about the simultaneous GH-congruence diagonalization of quaternionmatrices are obtained.展开更多
Let Q be the real quaternion field.Let the set of all matrices A=(α<sub>ij</sub>)<sub>n×m</sub>be Q<sub>n×m</sub>where α<sub>ij</sub>∈Q,A<sup>+</su...Let Q be the real quaternion field.Let the set of all matrices A=(α<sub>ij</sub>)<sub>n×m</sub>be Q<sub>n×m</sub>where α<sub>ij</sub>∈Q,A<sup>+</sup> be the conjugate transpose of A.Over a long period of time,there have been various kinds of definitions of determinantover the real quaternion field,but all of them are not concerning the entries of thematrix directly,that is"undirectly".From the point of view how the theory of determi-nant is algebraically developed for skew field,it may be worthwhile defined展开更多
In this paper,we shall prove that for any positive interger n,there exists non-trivialcommutative finite semigroup of idempotent consisting of some n×n real quaternion matri-ces which is lower semilattice.In the ...In this paper,we shall prove that for any positive interger n,there exists non-trivialcommutative finite semigroup of idempotent consisting of some n×n real quaternion matri-ces which is lower semilattice.In the process of solving this problem we shall see thatmany properties of generalized inverses for complex matrices still hold for quaternions ma-展开更多
This paper improves and generalizes famous Fischer's inequality and Hadamard's inequality,gets precise estimation of bounds for the determinant of quaternion matrix.
Let H be the real quaternion field,C and R be the complex and real field respectively.Clearly R（?）C（?）H. Let H<sup>m×n</sup> denote the set of all m×n matrices over H.If A=(a<sub>rs<...Let H be the real quaternion field,C and R be the complex and real field respectively.Clearly R（?）C（?）H. Let H<sup>m×n</sup> denote the set of all m×n matrices over H.If A=(a<sub>rs</sub>)∈H<sup>m×n</sup>,then there exist A<sub>1</sub> and A<sub>2</sub>∈C<sup>m×n</sup> such that A=A<sub>1</sub>+A<sub>2</sub>j.Let A<sub>C</sub> denote the complexrepresentation of A,that is the 2m×2n complex matrix Ac=（（A<sub>1</sub>/A<sub>2</sub>）（-A<sub>2</sub>/A<sub>1</sub>））（see[1,2]）.We denote by A<sup>D</sup> the Drazin inverse of A∈H<sup>m×n</sup> which is the unique solution of the e-展开更多
On the basis of the paoers[3—7],this paper study the monotonicity problems for the positive semidefinite generalized inverses of the positive semidefinite self-conjugate matrices of quaternions in the Lowner partial ...On the basis of the paoers[3—7],this paper study the monotonicity problems for the positive semidefinite generalized inverses of the positive semidefinite self-conjugate matrices of quaternions in the Lowner partial order,give the explicit formulations of the monotonicity solution sets A{1;≥,T_1;≤B^(1)}and B{1;≥,T_2≥A^(1)}for the(1)-inverse,and two results of the monotonicity charac teriaztion for the(1,2)-inverse.展开更多
设SC_n(Q)表示所有n×n的四元数自共轭矩阵的集合。当A∈SC_n(Q)时,λ_s(A)为A的特征值且满足λ_1(A)≥λ_2(A)≥…≥λ_n(A)。本文证明了 1)如果A∈SC_n(Q), P∈Q^(n×n) ,则 sum from s=1 to k(λ_(n-s-1)(PAP^n))≤sum from ...设SC_n(Q)表示所有n×n的四元数自共轭矩阵的集合。当A∈SC_n(Q)时,λ_s(A)为A的特征值且满足λ_1(A)≥λ_2(A)≥…≥λ_n(A)。本文证明了 1)如果A∈SC_n(Q), P∈Q^(n×n) ,则 sum from s=1 to k(λ_(n-s-1)(PAP^n))≤sum from s=1 to k(λ_s(A)λ_s(PP^n)), k=1,2,...,n; sum from s=1 to k(λ_(n-s+1)(A)λ_s(PP^n))≤sum from s=1 to k(λ_s(PAP^n)), 2) 如果A,B,C∈SC_n(Q)且B=C-A, 则sum from s=1 to k(λ_s^2(B))≥sum from s=1 to k ([λ_s(C)-λ_s(A)]~2).展开更多
文摘In this paper,the GH-congruence canonical forms of positive semidefinite and definte inite and definite(need not be self-conjugate)quaternion matrices are given,and a neccessary and sufficientcondition of GH-congruence for two positive semidifinite(definite)quaternion matrices isgiven also.Then simultaneous GH-congruence reduced forms for two self-conjugate matri-ces and some result about the simultaneous GH-congruence diagonalization of quaternionmatrices are obtained.
文摘Let Q be the real quaternion field.Let the set of all matrices A=(α<sub>ij</sub>)<sub>n×m</sub>be Q<sub>n×m</sub>where α<sub>ij</sub>∈Q,A<sup>+</sup> be the conjugate transpose of A.Over a long period of time,there have been various kinds of definitions of determinantover the real quaternion field,but all of them are not concerning the entries of thematrix directly,that is"undirectly".From the point of view how the theory of determi-nant is algebraically developed for skew field,it may be worthwhile defined
文摘In this paper,we shall prove that for any positive interger n,there exists non-trivialcommutative finite semigroup of idempotent consisting of some n×n real quaternion matri-ces which is lower semilattice.In the process of solving this problem we shall see thatmany properties of generalized inverses for complex matrices still hold for quaternions ma-
文摘This paper improves and generalizes famous Fischer's inequality and Hadamard's inequality,gets precise estimation of bounds for the determinant of quaternion matrix.
基金Supported by the Natural Science Foundation of jiangxi
文摘Let H be the real quaternion field,C and R be the complex and real field respectively.Clearly R（?）C（?）H. Let H<sup>m×n</sup> denote the set of all m×n matrices over H.If A=(a<sub>rs</sub>)∈H<sup>m×n</sup>,then there exist A<sub>1</sub> and A<sub>2</sub>∈C<sup>m×n</sup> such that A=A<sub>1</sub>+A<sub>2</sub>j.Let A<sub>C</sub> denote the complexrepresentation of A,that is the 2m×2n complex matrix Ac=（（A<sub>1</sub>/A<sub>2</sub>）（-A<sub>2</sub>/A<sub>1</sub>））（see[1,2]）.We denote by A<sup>D</sup> the Drazin inverse of A∈H<sup>m×n</sup> which is the unique solution of the e-
文摘On the basis of the paoers[3—7],this paper study the monotonicity problems for the positive semidefinite generalized inverses of the positive semidefinite self-conjugate matrices of quaternions in the Lowner partial order,give the explicit formulations of the monotonicity solution sets A{1;≥,T_1;≤B^(1)}and B{1;≥,T_2≥A^(1)}for the(1)-inverse,and two results of the monotonicity charac teriaztion for the(1,2)-inverse.
文摘设SC_n(Q)表示所有n×n的四元数自共轭矩阵的集合。当A∈SC_n(Q)时,λ_s(A)为A的特征值且满足λ_1(A)≥λ_2(A)≥…≥λ_n(A)。本文证明了 1)如果A∈SC_n(Q), P∈Q^(n×n) ,则 sum from s=1 to k(λ_(n-s-1)(PAP^n))≤sum from s=1 to k(λ_s(A)λ_s(PP^n)), k=1,2,...,n; sum from s=1 to k(λ_(n-s+1)(A)λ_s(PP^n))≤sum from s=1 to k(λ_s(PAP^n)), 2) 如果A,B,C∈SC_n(Q)且B=C-A, 则sum from s=1 to k(λ_s^2(B))≥sum from s=1 to k ([λ_s(C)-λ_s(A)]~2).
