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Let be field, and let A and B be n × n matrices with elements in . Suppose that A is completely reducible and that B is symmetric. If the principal minors of A determined by the 1- and 2-circuits of the graph of B and by the chordless circuits of the graph of A are equal to the corresponding principal minors of B, then A is diagonally similar to B; and conversely. 相似文献
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H.J. Ryser 《Discrete Mathematics》1984,49(2):169-173
Let A and B be (0, 1)-matrices of sizes m by t and t by n, respectively. Let x1, …, xt denote t independent indeterminates over the rational field Q and define X = diag[xt, …, xt]. We study the matrix equation AXB = Y. We first discuss its combinatorial significance relative to topics such as set intersections and the Marica-Schönheim theorem on set differences. We then prove the following theorem concerning the matrix Y. Suppose that the matrix Y of size m by n has rank m. Then Y contains m distinct nonzero elements, one in each of the m rows of Y. 相似文献
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The incidence matrix of a (υ, k, λ)-design is a (0, 1)-matrix A of order υ that satisfies the matrix equation AAT=(k?λ)I+λJ, where AT denotes the transpose of the matrix A, I is the identity matrix of order υ, J is the matrix of 1's of order υ, and υ, k, λ are integers such that 0<λ<k<υ?1. This matrix equation along with various modifications and generalizations has been extensively studied over many years. The theory presents an intriguing joining together of combinatorics, number theory, and matrix theory. We survey a portion of the recent literature. We discuss such varied topics as integral solutions, completion theorems, and λ-designs. We also discuss related topics such as Hadamard matrices and finite projective planes. Throughout the discussion we mention a number of basic problems that remain unsolved. 相似文献
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We prove that the union of a Riesz set and a Lust-Piquard set is a Riesz set. This gives as corollaries known results of Y. Katznelson, R.E. Dressler-L. Pigno, and D. Li. Moreover, we give an example of a Rosenthal set which is dense in Z for the Bohr topology. 相似文献
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Let $\mathcal X $ and $\mathcal Y $ be Banach spaces, and let $A\in \mathcal B (\mathcal X )$ and $C\in \mathcal B (\mathcal Y , \mathcal X )$ be given operators. A necessary and sufficient condition is given for $\left[ \begin{array}{cc} A&C \\ X&Y \\ \end{array} \right]$ to be invertible (respectively, left invertible) for some $X\in \mathcal B (\mathcal X , \mathcal Y )$ and $Y\in \mathcal B (\mathcal Y )$ . Furthermore, some related results are obtained. 相似文献
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Tu Boxun 《数学年刊B辑(英文版)》1982,3(2):249-259
Let \Omega be a field, and let F denote the Frobenius matrix:
$[F = \left( {\begin{array}{*{20}{c}}
0&{ - {\alpha _n}}\{{E_{n - 1}}}&\alpha
\end{array}} \right)\]$
where \alpha is an n-1 dimentional vector over Q, and E_n- 1 is identity matrix over \Omega.
Theorem 1. There hold two elementary decompositions of Frobenius matrix:
(i) F=SJB,
where S, J are two symmetric matrices, and B is an involutory matrix;
(ii) F=CQD,
where O is an involutory matrix, Q is an orthogonal matrix over \Omega, and D is a
diagonal matrix.
We use the decomposition (i) to deduce the following two theorems:
Theorem 2. Every square matrix over \Omega is a product of twe symmetric matrices
and one involutory matrix.
Theorem 3. Every square matrix over \Omega is a product of not more than four
symmetric matrices.
By using the decomposition (ii), we easily verify the following
Theorem 4(Wonenburger-Djokovic') . The necessary and sufficient condition
that a square matrix A may be decomposed as a product of two involutory matrices is
that A is nonsingular and similar to its inverse A^-1 over Q (See [2, 3]).
We also use the decomosition (ii) to obtain
Theorem 5. Every unimodular matrix is similar to the matrix CQB, where
C, B are two involutory matrices, and Q is an orthogonal matrix over Q.
As a consequence of Theorem 5. we deduce immediately the following
Theorem 6 (Gustafson-Halmos-Radjavi). Every unimodular matrix may be
decomposed as a product of not more than four involutory matrices (See [1] ).
Finally, we use the decomposition (ii) to derive the following
Thoerem 7. If the unimodular matrix A possesses one invariant factor which
is not constant polynomial, or the determinant of the unimodular matrix A is I and
A possesses two invariant factors with the same degree (>0), then A may be
decomposed as a product of three involutory matrices.
All of the proofs of the above theorems are constructive. 相似文献
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本文重点讨论了布尔矩阵半群中的幂等、广义幂等、素矩阵以及它们之间的联系,对非负矩阵半群上的类零型结构也进行了研究. 相似文献
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在本文中,我们证明了对一个反Krylov矩阵作QR分解后,利用得到的正交矩阵可以将一个具有互异特征值的对称矩阵转化为一个半可分矩阵的形式,这个结果表明了反Krylov矩阵与半可分矩阵之间的联系.另外,我们还证明了这类对称半可分矩阵在QR达代下矩阵结构保持不变性. 相似文献
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