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Square matrix

About: Square matrix is a research topic. Over the lifetime, 5000 publications have been published within this topic receiving 92428 citations.


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01 Jan 1984
TL;DR: In this article, the stability of self-adjoint rational matrix functions and matrix polynomials, as well as hermitian solutions of symmetric matrix algebraic Riccati equations, is studied.
Abstract: The stability of various factorizations of self-adjoint rational matrix functions and matrix polynomials, as well as of hermitian solutions of symmetric matrix algebraic Riccati equations, is studied. In the first part of this paper results on stability of certain classes of invariant subspaces of a matrix which is self-adjoint in an indefinite inner product were obtained. These results serve as the main tools in the investigation.

21 citations

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TL;DR: The notion of the symmetrized determinant was introduced in this paper for a fixed finite-dimensional associative algebra, and it was shown that for any fixed-dimensional algebra, the determinant of a matrix with entries in the algebra can be computed in polynomial time.
Abstract: We introduce a new notion of the determinant, called symmetrized determinant, for a square matrix with the entries in an associative algebra $\AA$. The monomial expansion of the symmetrized determinant is obtained from the standard expansion of the commutative determinant by averaging the products of entries of the matrix in all possible orders. We show that for any fixed finite-dimensional associative algebra $\AA$, the symmetrized determinant of an $n\times n$ matrix with the entries in $\AA$ can be computed in polynomial in $n$ time (the degree of the polynomial is linear in the dimension of $\AA$). Then, for every associative algebra $\AA$ endowed with a scalar product and unbiased probability measure, we construct a randomized polynomial time algorithm to estimate the permanent of non-negative matrices. We conjecture that if $\AA=\Mat(d, {\Bbb R})$ is the algebra of $d\times d$ real matrices endowed with the standard scalar product and Gaussian measure, the algorithm approximates the permanent of a non-negative $n \times n$ matrix within $O(\gamma_d^n)$ factor, where $\lim_{d \longrightarrow +\infty} \gamma_d=1$. Finally, we provide some informal arguments why the conjecture might be true.

21 citations

Journal ArticleDOI
TL;DR: In this article, the Weyl surface describing the dependence of Green's matrix on the boundary conditions is interpreted as the set of maximally isotropic subspaces of a quadratic form given by the Wronskian.
Abstract: A Jacobi matrix with matrix entries is a self-adjoint block tridiagonal matrix with invertible blocks on the off-diagonals. The Weyl surface describing the dependence of Green’s matrix on the boundary conditions is interpreted as the set of maximally isotropic subspaces of a quadratic form given by the Wronskian. Analysis of the possibly degenerate limit quadratic form leads to the limit point/limit surface theory of maximal symmetric extensions for semi-infinite Jacobi matrices with matrix entries with arbitrary deficiency indices. The resolvent of the extensions is calculated explicitly.

21 citations

Journal ArticleDOI
TL;DR: A formula for the inverse of any nonsingular matrix partitioned into two-by-two blocks is derived through a decomposition of the matrix itself and generalized inverses of the submatrices in the matrix.
Abstract: A formula for the inverse of any nonsingular matrix partitioned into two-by-two blocks is derived through a decomposition of the matrix itself and generalized inverses of the submatrices in the matrix. The formula is then applied to three matrix inverse completion problems to obtain their complete solutions.

21 citations

Journal ArticleDOI
TL;DR: It is shown that the best approximation is unique and provide an expression for this nearest matrix which is nearest to A^* in the Frobenius norm.

20 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202322
202244
2021115
2020149
2019134
2018145