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Enumerating partial linear transformations in a similarity class
- Source :
- Linear Algebra and its Applications. 625:196-211
- Publication Year :
- 2021
- Publisher :
- Elsevier BV, 2021.
-
Abstract
- Let $V$ be a finite-dimensional vector space over the finite field ${\mathbb F}_q$ and suppose $W$ and $\widetilde{W}$ are subspaces of $V$. Two linear transformations $T:W\to V$ and $\widetilde{T}:\widetilde{W}\to V$ are said to be similar if there exists a linear isomorphism $S:V\to V$ with $SW=\widetilde{W}$ such that $S\circ T=\widetilde{T}\circ S $. Given a linear map $T$ defined on a subspace $W$ of $V$, we give an explicit formula for the number of linear maps that are similar to $T$. Our results extend a theorem of Philip Hall that settles the case $W=V$ where the above problem is equivalent to counting the number of square matrices over ${\mathbb F}_q$ in a conjugacy class.<br />15 pages, 3 figures
- Subjects :
- Numerical Analysis
Algebra and Number Theory
Similarity (geometry)
010102 general mathematics
010103 numerical & computational mathematics
01 natural sciences
Linear subspace
Square matrix
Combinatorics
Linear map
Finite field
Conjugacy class
FOS: Mathematics
Mathematics - Combinatorics
Discrete Mathematics and Combinatorics
Combinatorics (math.CO)
Geometry and Topology
0101 mathematics
05A05, 05A10, 15B33
Subspace topology
Mathematics
Vector space
Subjects
Details
- ISSN :
- 00243795
- Volume :
- 625
- Database :
- OpenAIRE
- Journal :
- Linear Algebra and its Applications
- Accession number :
- edsair.doi.dedup.....b07681fc597f2e4515f1ddbc193ab343