2
votes
2answers
288 views

the spectrum of matrix with positive entries

It is well known that a matrix which all entries are positive real numbers, has a positive eigenvalue.(see algebraic topology, by Allen Hatcher). Now is the following generalization, true? Let A ...
2
votes
2answers
218 views

Are the finite dimensional von Neumann algebras, singly generated?

Let $\mathcal{M}$ be a finite dimensional von Neumann algebra, then : $$\mathcal{M} \simeq \bigoplus_i M_{n_i}(\mathbb{C})$$ Question : Is it singly generated (as von Neumann algebra)? how ? ...
5
votes
0answers
155 views

Bound on number of multiplications required to generate a matrix algebra from generators?

I've got a question about matrices and matrix algebras that offhand seems difficult, I'm wondering there is any sharp solution? Or perhaps it's known to not have any solution at all? Suppose you have ...
12
votes
0answers
318 views

Noncommutative arithmetic mean geometric mean inequality and symmetric polynomials

While analyzing convergence speed of stochastic-gradient methods for convex optimization problems, Recht et al (2011) posed a tantalizing conjecture. It seems quite tricky, so after having struggled a ...
4
votes
2answers
243 views

system of homogeneous matrix equations

Edit: Maybe I should make it clear that by a solution I mean a pair of matrices $(A,B)$ such that the identity below holds for any complex numbers $x,y$. One of my friend asked me the following ...
1
vote
0answers
131 views

Norm bound of the entrywise logarithm of a stochastic matrix stationary matrix

Hello, Denote $\log_\star$ as the entrywise logarithm operation, and let $A$ be some row-stochastic matrix such that $\lim_{p\rightarrow\infty}A^p$ exists and all its entries are non-zero. As a part ...
3
votes
1answer
293 views

Eigenvalues of certain positive matrices

For a matrix $ Q = (q_{ij}) \in GL_n(\mathbb{C}) $ let $ \overline{Q} = (\overline{q_{ij}}) $ be the matrix obtained by entry-wise complex conjugation (equivalently, $ \overline{Q} $ is the ...
2
votes
1answer
381 views

about decomposition of a non-negative definite operators

Hello, Many years before, I had the following problem. We first give a definition. Given a non-negative definite real-valued definite matrix $n^2\times n^2$ matrix $M$, it is called separable if it ...
13
votes
4answers
820 views

Are almost commuting hermitian matrices close to commuting matrices (in the 2-norm)?

I consider on $M_n(\mathbb C)$ the normalized $2$-norm, i.e. the norm given by $\|A\|_2 = \sqrt{Tr(A^* A)/n}$. My question is whether a $k$-uple of hermitian matrices that are almost commuting (with ...
0
votes
3answers
666 views

Is there any conclusions generalized Singular Value Decomposition into Hilbert Space

Spectrum decomposition can be regarded as the generalizations of the following fact that: Every Hermitian matrix $A$ can be decomposed into $A=U^{*}\Lambda U$,where $U$ is a unitary matrix Singular ...
16
votes
1answer
1k views

Commuting unitaries

Is the following true: For every unit vectors $x_1,..., x_n$, $y_1,..., y_n$ in $\mathbb{C}^k$ there exist a Hilbert space $H$, unitary operators $U_1,...,U_n$ and $V_1,...,V_n$ in $B(H)$ and unit ...
2
votes
2answers
207 views

unitary reduction of $q$-normal matrices

The unitary reduction of normal matrices is a well-known fact: if $A\in M_n(\mathbb C)$ commutes with its Hermitian adjoint $A^*$, then there exists a unitary $U\in\mathbb U_n$ and a diagonal matrix ...
8
votes
0answers
590 views

Linear equations on unitary operators

Let $\alpha_1^{(s)},\dots, \alpha_n^{(s)} \in \mathbb{C}, \quad s=1, \ldots, r.$ Question: Are the following conditions equivalent: There are unitary operators $U_1, \ldots, U_n\in B(H)$ such ...
3
votes
0answers
284 views

A question about the generalized Lidskii-Wielandt inequality for matrices proved by Thompson and Freede

In 1971, Thomson and Freede generalized the Lidskii-Wielandt inequalites as follows (version for singular values) Let $A$, $B$ be $n\times n$ Hermitian matrices. Suppose $\alpha_1\geq \alpha_2 \geq ...
14
votes
2answers
668 views

“Explicit” embedding of $\ell^1$ as a closed subalgebra of a direct sum of matrix algebras

For sake of brevity let $A$ denote the Banach algebra formed by equipping $\ell^1({\mathbb N})$ with pointwise multiplication. This algebra is clearly not isomorphic as a Banach algebra to any uniform ...
4
votes
0answers
235 views

A matrix minimisation problem

Feel free to edit the title! Suppose A is a C*-algebra and $a,b\in A$ are self-adjoint. I'd be very happy with A being just $n\times n$ matrices. Question: If there are $t\in\mathbb R$ and ...
5
votes
1answer
555 views

Orthogonal similarity of matrices

Given $M\in M_n({\mathbb R})$ and $\ell\in{0,\ldots,n-1}$, we define $$d_\ell(M)=\sum_{j=1}^nm_{j,j+\ell},$$ where the indices are understood mod $n$. In particular, $d_0$ is the trace operator. Let ...
2
votes
0answers
157 views

The orthogonal of $[A,B]$ in $M_n(k)$

Let ${\mathcal A}$ be the algebra spanned by the words in two letters $x$ and $y$. Its (infinite) basis is $1,x,y,x^2,xy,yx,y^2,...$ Let ${\mathcal A}_0$ be the sub-space (warning: not the ...
14
votes
1answer
554 views

Standard polynomials applied to matrices

The standard polynomial in $r$ non-commuting indeterminates $x_1,\ldots,x_r$ is defined by $${\mathcal S}_r(x_1,\ldots,x_r):=\sum_{\sigma\in S_r}\epsilon(\sigma)x_{\sigma(1)}x_{\sigma(2)}\cdots ...
20
votes
2answers
2k views

Finite subgroups of unitary groups

Let $n$ be an integer. Camille Jordan showed that there exists some $m \in {\mathbb N}$ (depending on $n$), such that for any pair of $n \times n$-unitaries $u,v \in U(n)$ which generate a finite ...
53
votes
2answers
4k views

Norms of Commutators

If an $n$ by $n$ complex matrix $A$ has trace zero, then it is a commutator, which means that there are $n$ by $n$ matrices $B$ and $C$ so that $A= BC-CB$. What is the order of the best constant ...