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4 votes
1 answer
204 views

Making Hermitian matrices almost commute

Consider two Hermitian matrices $A, B \in \mathbb{C}^{n \times n}$. I'm interested in finding another Hermitian matrix $A'$ that is close to $A$ and almost commutes with $B$. More precisely, I'd like ...
21 votes
1 answer
2k views

Almost commuting unitary matrices

Suppose that $A_1,\dots, A_k$ are unitary matrices such that any two of them can be approximated by commuting unitary matrices. i.e. for any $i$ and $j$, there are unitary matrices $A_i'$ and $A_j'$ ...
8 votes
1 answer
353 views

$E_n(\ell^\infty)=SL_n(\ell^\infty)$?

Let $R$ be a commutative unital ring $R$ with unit element $1$. For $n\in \mathbb{N}=\{1,2,3,\cdots\}$, let $SL_n(R)$ be the group of all $n\times n$ matrices with entries from $R$ having ...
1 vote
2 answers
289 views

Any analysis on phase of eigenvalue of unitary matrix?

I understand that there are invariant Haar measure for eigenvalues of unitary matrix. I further understand that absolute value of eigenvalues of unitary matrix is 1. But, I could not find any analysis ...
5 votes
0 answers
148 views

Groups of operators between local unitaries and full unitaries

Consider the group $U(d_1) \otimes U(d_2)$ of "local unitary" operators acting on the complex space $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$ (i.e., $U(d_1) \otimes U(d_2)$ is the group of unitary ...
1 vote
1 answer
387 views

All and the only algebraically closed fields s.t. any regular n-by-n matrix has a k-th root for every k

The title has it all. I'm looking for a proof/disproof of the fact that an algebraically closed field, say $\mathbb K$, has characteristic zero iff the following property (R) holds: For all $n,k \in \...