On families of anticommuting matrices

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• 作者：Pavel Hrube&scaron ; ^{pahrubes@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：15A-99
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：15 March 2016
• 年：2016
• 卷：493
• 期：Complete
• 页码：494-507
• 全文大小：348 K

文摘

Let e₁,…,e_k$e_1,\dots ,e_k$ be complex n×n$n\times n$ matrices such that e_ie_j=−e_je_i$e_i{e}_j=-e_j{e}_i$ whenever 19c78a9781b65c01717d327" title="Click to view the MathML source">i≠j$i\ne j$. We conjecture that $\mathrm{rk}(e_1^2)+\mathrm{rk}(e_2^2)+\cdots +\mathrm{rk}(e_k^2)\le O(n\log n)$. We show that:

(i).

$\mathrm{rk}(e_1^n)+\mathrm{rk}(e_2^n)+\cdots +\mathrm{rk}(e_k^n)\le O(n\log n)$,

(ii).

if $e_1^2,\dots ,e_k^2\ne 0$ then c994a6b9" title="Click to view the MathML source">k≤O(n)$k\le O(n)$,

(iii).

if e₁,…,e_k$e_1,\dots ,e_k$ have full rank, or at least n−O(n/log⁡n)$n-O(n/\log n)$, then k≤O(log⁡n)$k\le O(\log n)$.

(i) implies that the conjecture holds if $e_1^2,\dots ,e_k^2$ are diagonalisable (or if e₁,…,e_k$e_1,\dots ,e_k$ are). (ii) and (iii) show it holds when their rank is sufficiently large or sufficiently small.