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]. / Science, Faculty of / Mathematics, Department of / Graduate
| Identifer | oai:union.ndltd.org:UBC/oai:circle.library.ubc.ca:2429/34721 |
| Date | January 1970 |
| Creators | Kapoor, Jagmohan |
| Publisher | University of British Columbia |
| Source Sets | University of British Columbia |
| Language | English |
| Detected Language | English |
| Type | Text, Thesis/Dissertation |
| Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
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