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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 3cycles {(r, r+1, t) ; r = 1, …, n2; t = r+2, …, n} and let I denote the identity matrix in Sn. We shall denote the nlst 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 kdistinct real numbers. Determine the set of nsquare matrices A such that {tr(PA):P є H} = {a1, ..., ak} . We examine the cases when (i) H = Sn, k = 1 / (ii) H = {2cycles 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 skewsymmetric matrices and is of dimension [formula omitted]. Let A be an nsquare 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 2cycles (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 n1 nsquare 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

Creator  
Publisher 
University of British Columbia

Date Issued 
1970

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 3cycles {(r, r+1, t) ; r = 1, …, n2; t = r+2, …, n} and let
I denote the identity matrix in Sn. We shall denote the nlst
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 kdistinct real numbers. Determine the set of nsquare matrices A such that {tr(PA):P є H} = {a1, ..., ak} . We examine the cases
when
(i) H = Sn, k = 1 /
(ii) H = {2cycles 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 skewsymmetric matrices and is of dimension [formula omitted].
Let A be an nsquare 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 2cycles (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 n1 nsquare 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].

Genre  
Type  
Language 
eng

Date Available 
20110519

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0080476

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

Item Media
Item Citations and Data
Rights
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.