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1.
A method to characterize the class of all generalized inverses of any given matrix A is considered. Given a matrix A and a nonsingular bordered matrix T of A, the submatrix, corresponding to A, of T-1 is a generalized inverse of A, and conversely, any generalized inverse of A is obtainable by this method. There are different definitions of a generalized inverse, and the arguments are developed with the least restrictive definition. The characterization of the Moore-Penrose inverse, the most restrictive definition, is also considered. 相似文献
2.
Let A be a real symmetric n × n matrix of rank k, and suppose that A = BB′ for some real n × m matrix B with nonnegative entries (for some m). (Such an A is called completely positive.) It is shown that such a B exists with , where 2N is the maximal number of (off-diagonal) entries which equal zero in a nonsingular principal submatrix of A. An example is given where the least m which works is (k odd), (k even). 相似文献
3.
The Ostrowski-Reich theorem states that for a system Ax =b of linear equations with A nonsingular, if A is hermitian and if the diagonal of A is positive, then the SOR method converges for each relaxation parameter in (0,2) if and only if A is positive definite. This is actually a special case of the Householder-John theorem, which states that for A=M?N with A,M nonsingular, if A is hermitian and is positive definite, then M?1N is a convergent matrix if and only if A is positive definite. Our purposes here are to generalize the Householder-John theorem and to provide an insight into how and why the SOR method can converge. As a result the Ostrowski-Reich theorem is extended in two directions; one is when A is hermitian but the diagonal of A is not necessarily positive, so that A is not necessarily positive definite, and the other is when is positive definite but A is not necessarily hermitian. In the process, several other convergence results are obtained for general splittings of A. However, no claims are made concerning the case in which the convergence results obtained here can be applied to practical situations. 相似文献
4.
Oscar H Ibarra Shlomo Moran Roger Hui 《Journal of Algorithms in Cognition, Informatics and Logic》1982,3(1):45-56
We show that any m × n matrix A, over any field, can be written as a product, LSP, of three matrices, where L is a lower triangular matrix with l's on the main diagonal, S is an m × n matrix which reduces to an upper triangular matrix with nonzero diagonal elements when the zero rows are deleted, and P is an n × n permutation matrix. Moreover, L, S, and P can be found in O(mα?1n) time, where the complexity of matrix multiplication is O(mα). We use the LSP decomposition to construct fast algorithms for some important matrix problems. In particular, we develop O(mα?1n) algorithms for the following problems, where A is any m × n matrix: (1) Determine if the system of equations (where is a column vector) has a solution, and if so, find one such solution. (2) Find a generalized inverse, , of A (i.e., ). (3) Find simultaneously a maximal independent set of rows and a maximal independent set of columns of A. 相似文献
5.
The square roots of a complex n×n matrix A for which the real part of eixA, where , is positive definite are investigated. It is shown, for example, that when (i.e., A is strictly dissipative), A has a unique square root whose real and imaginary parts are both positive definite. 相似文献
6.
The concepts of matrix monotonicity, generalized inverse-positivity and splittings are investigated and are used to characterize the class of all M-matrices A, extending the well-known property that A?1?0 whenever A is nonsingular. These conditions are grouped into classes in order to identify those that are equivalent for arbitrary real matrices A. It is shown how the nonnegativity of a generalized left inverse of A plays a fundamental role in such characterizations, thereby extending recent work by one of the authors, by Meyer and Stadelmaier and by Rothblum. In addition, new characterizations are provided for the class of M-matrices with “property c”; that is, matrices A having a representation A=sI?B, s>0, B?0, where the powers of converge. Applications of these results to the study of iterative methods for solving arbitrary systems of linear equations are given elsewhere. 相似文献
7.
Let be a real or complex n × n interval matrix. Then it is shown that the Neumann series is convergent iff the sequence {k} converges to the null matrix , i.e., iff the spectral radius of the real comparison matrix constructed in [2] is less than one. 相似文献
8.
W.M Oliva 《Journal of Differential Equations》1983,49(3):453-472
Using a Poincaré compactification, the linear homogeneous system of delay equations {x = Ax(t ? 1) (A is an n × n real matrix) induces a delay system π(A) on the sphere Sn. The points at infinity belong to an invariant submanifold Sn ? 1 of Sn. For an open and dense set of 2 × 2 matrices A with distinct eigenvalues, the system π(A) has only hyperbolic critical points (including the critical points at infinity). For an open and dense set of 2 × 2matrices with complex eigenvalues, the nonwandering set at infinity is the union of an odd number of hyperbolic periodic orbits; if , the restriction of to S1 is Morse-Smale. For n = 1 there exist periodic orbits of period 4 provided that and Hopf bifurcation of a center occurs for ?A near . 相似文献
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10.
Let A be a real square matrix, and let J? be an interval not containing an eigenvalue of A. Is A–D nonsingular for all diagonal matrices D with entries di ∈J? This holds if A is symmetric, but is not true in general. We prove a necessary condition and indicate implications for an equation with a diagonal field. 相似文献
11.
Let A be a minimizing matrix for the permanent over the face of Ωn determined by a fully indecomposable matrix. It is shown that A is fully indecomposable and positive elements of A have permanental minors equal to per(A). Furthermore a zero entry of A has its permanental minor greater than or equal to per(A), provided that same element of the face has its corresponding entry positive. For 2?n?9 the minimum value of the permanent of a nearly decomposable is . 相似文献
12.
Recently Magnus and Neudecker [3] derived the dispersion matrix of vec X′X, when X′ is a p × n random matrix (n > p) and vec X′ has the distribution . This note is concerned with the matrix quadratic form X′AX, where X′ is a defined above and A is a nonrandom (not necessarily symmetric) matrix. The dispersion matrix of vec X′AX is then derived by applying results of Magnus and Neudecker [3] and Neudecker and Wansbeek [4]. This generalizes an earlier result of Giguère and Styan [2] which assumes a symmetric A. 相似文献
13.
The Schur product of two n×n complex matrices A=(aij), B=(bij) is defined by A°B=(aijbij. By a result of Schur [2], the algebra of n×n matrices with Schur product and the usual addition is a commutative Banach algebra under the operator norm (the norm of the operator defined on n by the matrix). For a fixed matrix A, the norm of the operator B?A°B on this Banach algebra is called the Schur multiplier norm of A, and is denoted by ∥A∥m. It is proved here that for all unitary U (where ∥·∥ denotes the operator norm) iff A is a scalar multiple of a unitary matrix; and that ∥A∥m=∥A∥ iff there exist two permutations P, Q, a p×p (1?p?n) unitary U, an (n?p)×(n?p)1 contraction C, and a nonnegative number λ such that and this is so iff , where ā is the matrix obtained by taking entrywise conjugates of A. 相似文献
14.
It has been conjectured that if A is a doubly stochastic n>× n matrix such that per A(i, j)≥perA for all i, j, then either A = Jn, then n × n matrix with each entry equal to , or, up to permutations of rows and columns, , where Pn is a full cycle permutation matrix. This conjecture is proved. 相似文献
15.
Let A be a subalgebra of the full matrix algebra Mn(F), and suppose J∈A, where J is the Jordan block corresponding to xn. Let be a set of generators of A. It is shown that the graph of determines whether A is the full matrix algebra Mn(F). 相似文献
16.
Some techniques for the study of the algebraic curve C(A) which generates the numerical range W(A) of an n×n matrix A as its convex hull are developed. These enable one to give an explicit point equation of C(A) and a formula for the curvature of C(A) at a boundary point of W(A). Applied to the case of a nonnegative matrix A, a simple relation is found between the curvature of the function Φ(A)=p((1?α)A+ αAT) (pbeingthePerronroot) at and the curvature of W(A) at the Perron root of . A connection with 2-dimensional pencils of Hermitian matrices is mentioned and a conjecture formulated. 相似文献
17.
Let Mm,n(F) denote the space of all mXn matrices over the algebraically closed field F. A subspace of Mm,n(F), all of whose nonzero elements have rank k, is said to be essentially decomposable if there exist nonsingular mXn matrices U and V respectively such that for any element A, UAV has the form where A1 is iX(k–i) for some i?k. Theorem: If is a space of rank k matrices, then either is essentially decomposable or dim ?k+1. An example shows that the above bound on non-essentially-decomposable spaces of rank k matrices is sharp whenever n?2k–1. 相似文献
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20.
George Phillip Barker 《Linear algebra and its applications》1977,16(3):233-235
Let A be an n×n matrix with complex entries. A necessary and sufficient condition is established for the existence of a Hermitian solution H to the equations . 相似文献