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 Rank preservers on certain symmetry classes of tensors
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Rank preservers on certain symmetry classes of tensors Lim, MingHuat
Abstract
Let U denote a finite dimensional vector space over an algebraically closed field F . In this thesis, we are concerned with rank one preservers on the r(th) symmetric product spaces r/VU and rank k preservers on the 2nd Grassmann product spaces 2/AU. The main results are as follows: (i) Let T : [formula omitted] be a rank one preserver. (a) If dim U ≥ r + 1 , then T is induced by a nonsingular linear transformation on U . (This was proved by L.J. Cummings in his Ph.D. Thesis under the assumption that dim U > r + 1 and the characteristic of F is zero or greater than r .) (b) If 2 < dim U < r + 1 and the characteristic of F is zero or greater than r, then either T is induced by a nonsingular linear transformation on U or [formula omitted] for some two dimensional subspace W of U. (ii) Let [formula omitted] be a rank one preserver where r < s. If dim U ≥ s + 1 and the characteristic of F is zero or greater than s/r, then T is induced by s  r nonzero vectors of U and a nonsingular linear transformation on U. (iii) Let T : [formula omitted] be a rank k preserver and char F ≠ 2. If T is nonsingular or dim U = 2k or k = 2 , then T is a compound, except when dim U = 4 , in which case T may be the composite of a compound and a linear transformation induced by a correlation of the two dimensional subspaces of U.
Item Metadata
Title  Rank preservers on certain symmetry classes of tensors 
Creator  Lim, MingHuat 
Publisher  University of British Columbia 
Date Issued  1971 
Description 
Let U denote a finite dimensional vector space over an algebraically
closed field F . In this thesis, we are concerned with rank one
preservers on the r(th) symmetric product spaces r/VU and rank k preservers on the 2nd Grassmann product spaces 2/AU.
The main results are as follows:
(i) Let T : [formula omitted] be a rank one preserver.
(a) If dim U ≥ r + 1 , then T is induced by a nonsingular linear transformation on U . (This was proved by L.J. Cummings in his Ph.D. Thesis under the assumption that dim U > r + 1 and the characteristic of F is zero or greater than r .)
(b) If 2 < dim U < r + 1 and the characteristic of F is
zero or greater than r, then either T is induced by a nonsingular linear transformation on U or [formula omitted] for some two dimensional subspace W of U.
(ii) Let [formula omitted] be a rank one preserver where r < s.
If dim U ≥ s + 1 and the characteristic of F is zero or greater than s/r, then T is induced by s  r nonzero vectors of U and a nonsingular linear transformation on U. (iii) Let T : [formula omitted] be a rank k preserver and char F ≠ 2. If T is nonsingular or dim U = 2k or k = 2 , then T is a compound, except when dim U = 4 , in which case T may be the composite of a compound and a linear transformation induced by a correlation of the two dimensional subspaces of U.

Subject  Calculus of tensors 
Genre  Thesis/Dissertation 
Type  Text 
Language  eng 
Date Available  20110503 
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  10.14288/1.0302216 
URI  
Degree  Doctor of Philosophy  PhD 
Program  Mathematics 
Affiliation  Science, Faculty of; Mathematics, Department of 
Degree Grantor  University of British Columbia 
Campus  UBCV 
Scholarly Level  Graduate 
Aggregated Source Repository  DSpace 
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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.