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Matrices which, under row permutations, give specified values of certain matrix functions Kapoor, Jagmohan
Abstract
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].
Item Metadata
Title |
Matrices which, under row permutations, give specified values of certain matrix functions
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
1970
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Description |
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].
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Genre | |
Type | |
Language |
eng
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Date Available |
2011-05-19
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Provider |
Vancouver : University of British Columbia Library
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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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DOI |
10.14288/1.0080476
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URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Campus | |
Scholarly Level |
Graduate
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Aggregated Source Repository |
DSpace
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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.