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Matrices which, under row permutations, give specified values of certain matrix functions

University of British Columbia

Let Sn denote the set of n x n permutation matrices; let T denote the set of transpositions in Sn; let C denote the set of 3-cycles {(r, r+1, t) ; r = 1, …, n-2; t = r+2, …, n} and let I denote the identity matrix in Sn. We shall denote the n-lst elementary symmetric function of the eigenvalues of A by [formula omitted]. In this thesis, we pose the following problems: 1. Let H be a subset of Sn and a1, …, ak be k-distinct real numbers. Determine the set of n-square matrices A such that {tr(PA):P є H} = {a1, ..., ak} . We examine the cases when (i) H = Sn, k = 1 / (ii) H = {2-cycles in Sn} , k = 1 (iii) H = Sn, k = 2 2. Determine the set of n x. n matrices such that [formula omitted]. 3. Examine those orthogonal matrices which can be expressed as linear combinations of permutation matrices. The main results are as follows: If R’ is the subspace of rank 1 matrices with all rows equal and if C’ is the subspace of rank 1 matrices with all columns equal, then the n x n matrices A such that tr(PA) = tr(A) for all P є Sn form a subspace S = R' + C’.This implies- that the.' rank of A is ≤ 2. If tr(PA) = tr(A) for all P є T , then such A's form a subspace which contains all n x n skew-symmetric matrices and is of dimension [formula omitted]. Let A be an n-square matrix such that (tr(PA) : P є Sn} = {a1, a.2} , where a1 ≠ a2. Then A is either of the form C = A1 + A2, where A1 є (R’ +. C’) and A2 has entries a1 – a2 at [formula omitted], j =2, …, k and zeros elsewhere, or of the form CT. The set [formula omitted] consists of 1 n 1 all 2-cycles (r^, rj)> j = 2, ..., k and the products P of disjoint cycles [formula omitted], for which one of the P1 has its graph with an edge [formula omitted]. If A is rank n-1 n-square matrix with the property That[formula omitted] for all P є Sn, then A is of the form [formula omitted] where Ui are the row vectors. Finally,.if [formula omitted] , where all P1, are from an independent set TUCUI of Sn, is an orthogonal matrix, then [formula omitted].

Text

2011-05-19

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http://hdl.handle.net/2429/34721

Mathematics

University of British Columbia

Graduate