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On Simultaneous Similarity Of D-Tuples Of Commuting Square Matrices, Corey Connelly
On Simultaneous Similarity Of D-Tuples Of Commuting Square Matrices, Corey Connelly
USF Tampa Graduate Theses and Dissertations
It has been shown by B. Shekhtman that when any d-tuple A of pairwise commuting N × N matrices with complex entries is cyclic, then A is simultaneously similar to the d-tuple of commuting N × N matrices B if and only if B is cyclic, and the sets of polynomials in d variables which annihilate A and B are equivalent.
This thesis offers a further generalization of this result, demonstrating the necessary and sufficient conditions for the simultaneous similarity of n-cyclic d-tuples of commuting square complex-valued matrices.
Leonard Systems And Their Friends, Jonathan Spiewak
Leonard Systems And Their Friends, Jonathan Spiewak
USF Tampa Graduate Theses and Dissertations
Let $V$ be a finite-dimensional vector space over a field $\mathbb{K}$, and let
\text{End}$(V)$ be the set of all $\mathbb{K}$-linear transformations from $V$ to $V$.
A {\em Leonard system} on $V$ is a sequence
\[(\A ;\B; \lbrace E_i\rbrace_{i=0}^d; \lbrace E^*_i\rbrace_{i=0}^d),\]
where
$\A$ and $\B $ are multiplicity-free elements of \text{End}$(V)$;
$\lbrace E_i\rbrace_{i=0}^d$ and $\lbrace E^*_i\rbrace_{i=0}^d$
are orderings of the primitive idempotents of $\A $ and $\B$, respectively; and
for $0\leq i, j\leq d$, the expressions $E_i\B E_j$ and $E^*_i\A E^*_j$ are zero when $\vert i-j\vert > 1$ and
nonzero when $\vert i-j \vert = 1$.
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Leonard systems arise in connection …