Showing papers on "Square matrix published in 1969"

A Note on Kronecker Matrix Products and Matrix Equation Systems

Abstract: where B' is the transpose of B. It has been shown that Definition 1 and Theorem 1 can fruitfully be applied to problems of matrix differentiation [2]. In this note it will be shown that they can be applied to a more general class of linear matrix equations, including linear matrix differential equations. Firstly, four standard properties of Kronecker products have to be related, all of which may be proved in an elementary fashion [1, p. 223 if.]. The matrices involved can have any appropriate orders. In Property 4 it is assumed that A and B are square of order m and s, respectively. (The same order assumption will be made in Theorems 2 and 3, which will be presented further on.) PROPERTY 1. (A B)(C D) =(AC) (BD). PROPERTY 2. (A 0 B)' = A' 0 B'. PROPERTY. 3. (A + B) 09 (C + D) = A C + A D + B (& C + B X D. PROPERTY 4. If A has characteristic roots oci, i = 1, -*, m, and if B has characteristic roots /IB, j = 1,... , s, then A 0 B has characteristic roots oci,Bj. Further, Is 0 A + B 0) Im has characteristic roots oci + f3B. For the treatment of differential equations we need the matrix exponential

On Similarity and the Diagonal of a Matrix

Abstract: (1969). On Similarity and the Diagonal of a Matrix. The American Mathematical Monthly: Vol. 76, No. 2, pp. 167-169.

Orthogonalization Techniques of a Direction Cosine Matrix

Abstract: Three orthogonalization techniques to correct errors in the computeddirection cosine matrix are introduced. One of these techniques is avectorial technique based on the fact that the three rows of a directioncosine matrix constitute an orthonormal set of vectors in aree-threedimensional space. The other two iterative techniques are based onthe fact that the inverse and transpose of an orthogonal matrix areequal. In computing a time-varying direction cosine matrix computationalional errors are accompanied by the loss of the orthogonaliterty prop-rty of the matrix. When one of these three techniques is useo re-restore the orthogonality of the matrix, the computational errors arealso corrected. These techniques were tested experimentally and theresults, given in this paper, were compared with a method used by the Honeywell Corporation.

A recursive relation for the determinant of a pentadiagonal matrix

Abstract: A recursive relation, relating leading principal minors, is developed for the determinant of a pentadiagonal matrix. A numerical example is included to indicate its use in calculating eigenvalues.

Determinants, permanents and bipartite graphs,

Abstract: : The combinatorial properties of a nonnegative matrix M are captured by that binary matrix A = A(M) in which the entries are 1 whenever those of M are positive. If A is a square matrix, then it can be regarded as the adjacency matrix of a directed graph (digraph). If A is rectangular, a bipartite graph (bigraph) can be associated with A; of course this can also be done for A square. The determinant of the adjacency matrix of a graph or digraph has been expressed in terms of its structure, and so has the permanent. The purposes of this report are to express the permanent of a square or rectangular binary matrix in terms of the associated bigraph, and to formulate the determinant of a square matrix in terms of its bigraph. (Author)

Products of hermitian matrices and symmetries

Abstract: It is the main purpose of this note to prove that every complex matrix with real determinant is the product of four hermitian matrices; Theorem 2 is an actually stronger result. Every real square matrix is the product of two real hermitian matrices [1]; this is a special case of our Theorem 1 which is of interest in itself, if it is indeed new. Theorem 3 was motivated by a theorem of Halmos and Kakutani [3 ] who proved that every unitary operator on an infinitedimensional Hilbert space is the product of four symmetries (i.e., operators that are hermitian and unitary). We also show that the number of factors in these results cannot be reduced in general.

Semi-empirical all valence electrons SCF–MO–CNDO theory. Part 6.—Effective field gradient and diagonal of population matrix

Abstract: The principal axes of the electric field gradient are shown to be related to the diagonal elements of the charge density-bond order population matrix. Then the semi-empirical all valence electron complete neglect of differential overlap, SAVE–CNDO, theory is applied to PCl+4 and CCl4 applying the theory that, within the limitations of the CNDO theory, when the population submatrix is diagonal then the efg matrix is also diagonal.

On a Generalized Distribution of the Poles of the Unitary Collision Matrix

Abstract: An expression for the joint distribution of the complex poles of the unitary collision matrix is derived for the single‐channel case, which is valid for all values of the ratio of the width to the spacing. The derivation uses the statistical distribution of the parameters of the real R‐matrix theory. We find that unitarity gives rise to the statistical correlations between the width and the spacing of the collision matrix. It is shown that the distribution of the poles of the unitary collision matrix using Feshbach's unified theory of nuclear reactions is the same as the one obtained using R‐matrix theory, provided we make a particular choice of the arbitrary boundary condition in the latter theory. A remark is made about the use of the random complex orthogonal matrix in the study of the parameters of the statistical collision matrix.

The norm of a matrix after a similarity transformation

Abstract: The norm of a similarity transformation of a matrix is studied considered as a function of the transformation matrix, and conditions for stationary points of this function are given in terms of properties of the transformed matrix.

X.—Determinants for Matrices over Lattices

Abstract: In the literature concerning matrices whose co-ordinates are elements of a Boolean lattice, one may find three different definitions for the determinant of a matrix. We shall call these the first, second and third determinant and will denote the value of the i th determinant of a matrix A by | A | i for i = 1, 2, 3. The first determinant may be defined for square matrices over an arbitrary lattice. The second and third determinants may be defined for square matrices over any lattice L with a greatest element I, a least element o and an orthocomplementation′: L→L , that is a ′ is a complement of a, a = a″ and a ≤ b implies that b ′ ≤ a ′ for all a, b in L . In this paper we obtain some elementary properties of these determinants in this general setting and in the particular case where L is an orthomodular lattice, that is a lattice with o, 1 and an orthocomplementation' such that

Extremal properties of balanced tri-diagonal matrices

Abstract: If A is a square matrix with distinct eigenvalues and D a nonsingular matrix, then the angles between rowand column-eigenvectors of D-'AD differ from the corresponding quantities of A. Perturbation analysis of the eigenvalue problem motivates the minimization of functions of these angles over the set of diagonal similarity transforms; two such functions which are of particular interest are the spectral and the Euclidean condition numbers of the eigenvector matrix X of D-'AD. It is shown that for a tri-diagonal real matrix A both these condition numbers are minimized when D is chosen such that the magnitudes of corresponding suband super-diagonal elements are equal. * If a tri-diagonal matrix A is such that corresponding suband super-diagonal elements have equal magnitude then A is said to be balanced or equilibrated. Wilkinson [5, p. 424] uses norms of balanced tri-diagonal matrices for error analysis of the eigenvalue problem. He observes that, given a tri-diagonal matrix A = [aij] all of whose suband super-diagonal elements are nonzero, a diagonal matrix D = diag (d1, d2, * * *, dn) can be found such that D-1AD is balanced. In fact, such a D is defined by di+1di = (ai+i,i/Iaj,j+iI )1/2 i = 1, 2, *., n -1 If some subor super-diagonal element of A is zero then finding its eigenvalues can be reduced to finding the eigenvalues of submatrices, each of which can be balanced separately. It is an immediate consequence of Osborne's Lemma 2 [3] that a balanced tridiagonal matrix A has the extremal property JJAIIAE = inf ID 'ADJJE, D where 11 |IE denotes the Euclidean matrix norm (Schur norm, Frobenius norm). Our Theorem 1 states the analogous result for the spectral norm; Theorems 2 and 3 show that the eigenvalue problem of a balanced tri-diagonal matrix is optimally conditioned in the sense that no matrix of the form D-1AD has smaller angles between corresponding rowand column-eigenvectors. We use JJ 1 to denote the Euclidean vector norm, 11 * 112 for the subordinate matrix bound (the spectral mat?ix norm), k2(.) for the spectral condition number of a nonsingular matrix, and kE(*) for the Euclidean condition number (defined by kE(X) = IIXIIE IIX-111E). Absolute value signs applied to vectors are understood componentwise. D, D1, and D2 denote diagonal matrices with positive diagonal elements. THEOREM 1. If A is a balanced tri-diagonal real matrix then JA 112 = inf[JD-'AD112 D Received May 9, 1968.

On comparatively stable tridiagonalization methods

Abstract: A method for choosing certain matrices necessary for the "tridiagonalization" of an arbitrary, real, square matrix is sketched. As opposed to previous methods which choose modifications of matrices developed for other purposes, we choose these matrices on the basis of a direct examination of the essential computational problem occuring in "sequential tridiagonalization" and require that our choices have relatively small condition numbers.

Certification of algorithm 298 [F1]: determination of the square root of a positive definite matrix

Abstract: and an error bound E R R for V A L , i.e. I V A L -I [ _~ E R R . The integrand F ( X ) must be given as a function subprogram with the heading F U N C T I O N F ( X ) . V A L is obtained from a modified form of Romberg quadrature which is less sensit ive to the accumulation of rounding errors than the customary one. In this procedure, which was devised by Krasun and Prager [1], the following \"ske le ton\" Romberg table is constructed:

Subroutine Perm (Program Description)

Abstract: The algorithm computes the permanent of a square matrix which has been shown to yield the number of loops in a microwave network flow graph.

Transformation on a scalar-valued matrix product

Abstract: Some useful structures are derived for the scalar-valued matrix product x'Qy when substitutions x=Az and y=Bw are made. The value of such transformations is exemplified with a formulation for probabilistic parameter inaccuracy effects.