# Tagged Questions

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### Strong convexity of the trace of the square root of a matrix function

Any clues about how to prove that the following function is strongly-concave in $x$? (We conjecture it is $2$-strongly concave but cannot prove it. We have already proved strict concavity through ...
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### How can I prove that the negative biased triangular kernel is positive semidefinite

How can I prove that the following triangular kernel function defined in $[0, 1] \subset R^1$ $k(x, x') = (1 - 2|x-x'|)$ is a positive semidefinite function? It turns out to be psd function when ...
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### Alike looking matrices imply convergence of eigenvalues?

This is a question about convergence of eigenvalues which essentially came up in studying the spectrum of St.-Liouville operators. We want to look at matrices that agree in most of their entries and ...
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### Method to Generate Random Mutually Orthogonal Unitary Matrices

The title says it all, I'd like to know if such a method exists to generate random mutually orthogonal unitary matrices. If so, any supplementary references would be very much obliged. Thanks again!
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### Random variable matrix exponential

I am trying to find out the distribution of a matrix exponential which is a function of a random variable. My mathematics background is very limited and I hope I can receive some help from here. What ...
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### Schur product, partial order

Let $A, B$ be positive definite matrices. Then $A^r\circ B^r \le (A\circ B)^r$ for $0\le r\le 1$, where $\circ$ is Schur product. Here the inequality is in the sense of Loewner partial order. How to ...
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### 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 ...
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### Nth root of a matrix as an analytic function?

Let $A$ be a $k \times k$ invertible matrix over complex numbers. If it possible to write its nth root as an analytic function (i.e. power series in $A$)? EDIT: Complex coefficients can be functions ...
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### Second conjugate operators to operators on $c_0$

I posted my question at MS but unfortunately it is still without a response, so let me ask it here. We can think about a bounded operator $T\colon c_0\to c_0$ as a double-infinite matrix ...
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### Representing a product of matrix exponentials as the exponential of a sum

In Proof of a conjectured exponential formula, R. C. Thompson (1986) [edit: apparently, assuming Horn's conjecture] proved that if $A$ and $B$ are Hermitian matrices, then there exist unitary matrices ...
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### positive hermitian elements in $M_n(\mathbb{C})$

Elements of the set $P$ of positive hermitian $n×n$ matrices over complex numbers have some special properties: (i) they are closed under sum, (ii) they are closed under multiplication by positive ...
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### Characterizing invertible nonnegative matrices with bounded sums

Almost a year ago, I asked in this question about obtaining a tight bound on the sum of the entries of the inverse of a strictly positive definite matrix. Denis Serre gave a nice counterexample ...
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### A singular value inequality

Let $s_1,s_2: \mathbb{R}^{2\times 2} \mapsto \mathbb{R}_+$, $s_{1}\left(\cdot\right)\ge s_{2}\left(\cdot\right)\ge 0$, be the singular values of a $2\times2$ matrix. Is it true that ...
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### 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 ...
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### Subgroups of U(M_n)

can any subgroup of the unitary group of full matrix alg $M_d(\mathbb{C})$ be approximated on finite sets by a finite subgroup? i.e. is the following True or false? Let $n, d$ be positive integers ...
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### 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 ...
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### $\ell^p$ version of singular values

I am embarassed to pose this question. It is a generalization of a question asked less than 24 hours ago by an unknonw (Google), which has been deleted since then, presumably by its author ...
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### “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 ...
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### 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 ...
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### Random products of projections: bounds on convergence rate?

The von Neumann-Halperin [vN,H] theorem shows that iterating a fixed product of projection operators converges to the projector onto the intersection subspace of the individual projectors. A good ...
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### An analogue of an old proposition

For the absolute value $|C|=(C^*C)^\frac{1}{2}$ and the Hilbert-Schmidt norm $\parallel C\parallel_{HS}=(trC^*C)^\frac{1}{2}$ of the operator $C$. The following inequality is shown by Araki et al in ...
Let $M$ be a $3\times2$ matrix. Is it true that for any $x\in\mathbb{R}^{2}$ with $\left\Vert x\right\Vert _{3}=1$ there is some subspace $V$ with dimension $2$ of $\mathbb{R}^{3}$, such that ...
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 ...