Square matrix

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

The DAD theorem for arbitrary row sums

Abstract: Given an m x m symmetric nonnegative matrix A and a positive vector R = (rl, * , rm), necessary and sufficient conditions are obtained in order that there exist a diagonal matrix D with positive main diagonal such that DAD has row sum vector R. A nonnegative m x n matrix A is called completely decomposable if there exist partitions a1, a2 of 11, * mI and 01 02 of 11, ... , n} into nonvacuous sets such that A[a1, 02] and A[a2, Oil are zero matrices. Here we use the notation that A[a, /] is the submatrix of A whose rows are indexed by a and whose columns are indexed by /, the rows and columns in A[a, /] appearing in the same order as in A. If m = n, the matrix A is called completely reducible if there exists a partition a 1' a2 of II, **, ml into nonvacuous sets such that A[a,, a2] and A[a2, a1] are zero matrices. Generalizing theorems of Sinkhorn and Knopp [10] and Brualdi, Parter, and Schneider [1], Menon [7] proved the following theorem: Let A be an m x n nonnegative matrix and let R -= (r1, *. ., r ) and S = (s1, *. ., sn) be positive vectors with r1 + * + rm = s1 + --+ Sn. Let M(R, S) denote the class of all m x n nonnegative matrices with row sum vector R and column sum vector S. Then there exist diagonal matrices D 1 and D2 with positive main diagonals such that D 1AD2 is in 2(R, S) if and only if there is a matrix in 2(R, S) which has the same zero pattern as A. (We say that a matrix B has the same zero pattern as A provided bi= 0 if and only if ai = 0.) If, in addition, A is not completely decomposable, the diagonal matrices D1, D2 are unique up to positive scalar factor: if U1AU2 is in 2(R, S) then there exists & > 0 such that U1 = D Secondary 15A5 1.

Matrix functions and applications: Part IV — Matrix functions and constituent matrices

Abstract: This fourth part of a five-part series starts with the expansion of an analytic function of an arbitrary square matrix, involving real or complex numbers, in terms of its constituent matrices. The conjoint matrix and characteristic polynominal are computed by a convenient algorithm and are then used for calculating constituent matrices. Finally there is a discussion of the two-sided matrix equation AY + B = YC, and of the reduction of an arbitrary matrix to a similar Jordan matrix.

On the existence of square dot-matrix patterns having a specific three-valued periodic-correlation function

Abstract: The author examines square matrices of size n containing dot patterns satisfying the following two restrictions: (1) each column contain precisely one dot, and (2) if the pattern is moved around over a plane tied by the same pattern, when in all positions except the home position there is at most one overlap in dots. From differing viewpoints, there matrices are the characteristic functions of either a certain class of relative difference sets or else a select subset of bent functions. Also, the existence of such an (n*n) matrix implies the existence of a finite projective plane of order n. A family of constructions for such matrices is available when n is prime. A polynomial equation characterizing such matrices and resembling the Hall polynomial equation of cyclic difference sets is presented. Analogs of known existence tests for cyclic difference sets are then applied to rule out existence for most nonprime values of n. It is shown how such patterns can be used to provide hopping patterns for a frequency-hopped multiple-access system. >