6
votes
1answer
201 views

An inequality for positive definite matrices

Let $K$ and $K^\prime$ positive definite $n \times n$ matrices, such that for all vectors $f \ge 0$ with nonnegative coordinates we have $$\sum_{i,j} K_{ij} f_i f_j \le \sum_{ij} K^\prime_{ij} f_i ...
10
votes
1answer
209 views

Hölder's inequality for matrices

I was wondering if the Hölder's inequality was true for matrix induced norms, i.e. if $$\|AB\|_1 \leq \|A\|_p\|B\|_q, \quad\forall p,q \in [1,\infty] \text{ s.t. } \tfrac{1}{p}+\tfrac{1}{q} = 1.$$ But ...
5
votes
1answer
190 views

Given a correlation matrix $B$. What correlation matrix A (maximizes / minimizes) the following: det(A+B)

Given correlation matrix $B$ (positive semi-definite with ones in the diagonal). 1)Find the correlation matrix $A$ which maximizes $\det\left(A+B\right)$. 2)Find the correlation matrix $A$ which ...
4
votes
1answer
189 views

Determinant inequality of square-product sum of diagonal matrix and upper-triangular matrix

Recently, I have seen a matrix inequality but don't know how to prove it. The inequality goes as follows. For an arbitrary $n\times n$ diagonal matrix $\mathbf{D}$ and an arbitrary upper-triangular ...
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 ...
16
votes
2answers
389 views

What is $A+A^T$ when $A$ is row-stochastic ?

This is motivated by this MO question. If $A\in{\bf M}_n({\mathbb R})$ is row-stochastic (entrywise non-negative, and $\sum_j a_{ij}=1$ for all $i$), then $M:=A+A^T$ is symmetric, entrywise ...
3
votes
2answers
113 views

a monotone relation for s-numbers

Assume $A, B$ are self-ajoint compact operators. Is it true that $\|A+iB\|\le \|2A+iB\|$? Do we have a stronger inequality $\prod_{k=1}^ns_k(A+iB)\le \prod_{k=1}^ns_k(2A+iB)$ or even stronger one ...
0
votes
1answer
261 views

How to solve this optimization with the orthogonal constraint?

Problem Supposing that $A$ is a symmetric real matrix and $\{\mathbf{w}\_i\}_{i=1}^n$ is any orthogonal basis on $\mathbb{R}^n$ such that $W^\top W=WW^\top=\mathbf{I}_n$ where ...
5
votes
1answer
373 views

Trace matrix inequality

Hello all, I come across the following problem. Is it true that for a positive definite matrix $X^{n\times n}$, the following holds $\text{trace}(X^{-1})\geq\text{trace}([\text{diag}(X)]^{-1})$, ...
5
votes
1answer
392 views

Hadamard-like inequalites for positive definite symmetric matrices

Let $S$ be any positive semi-definite symmetric matrix (Hermitian psd matrices work as well). The Hadamard inequality is that $$\det S\le\prod_{i=1}^n s_{ii}.$$ My question is whether there are some ...
1
vote
1answer
151 views

Unitary matrix and matrix inequality [closed]

Dear all, Suppose U and V are unitary matrix, A and B are positive definite, Does: $UAU^{-1} < VBV^{-1}$ implies $A< B$ and vice versa?
4
votes
1answer
300 views

Is it possible to extend this inequality about Euclidean distance &Frobenius norm to more general Bregman divergence such as relative entropy & von Neumann divergence?

Motivation- A Special Case Supposing $A,B\in\mathbb{S}^{m\times m}$ are symmetric positive semi-definite (SPD) matrices and $\mathbf{x}\in\mathbb{R}^m$ is a unit vector where $\|\mathbf{x}\|=1$, we ...
3
votes
1answer
694 views

Ratio sum comparison on operators

It is known by the Lidskii inequality, that $\sum_{i=1}^n \left|s_i(S)-s_i(T)\right|\le\sum_{i=1}^n s_i(S+T)$, where $s_i(S)$ is the $i$-th singular value of $S$. How would one prove that ...
1
vote
1answer
445 views

A question on gauge functions

In the second paragraph on Page 71 of the book Matrix Analysis by Bhatia, 1997, it says ``as a consequence of (III.12) we have Theorem III 4.4''. How can one get the inequality in Theorem III 4.4 from ...
1
vote
1answer
694 views

On an eigenvalue inequality

Let $\lambda_1 (\cdot)$ be the larger absolute value eigenvalue of a $2\times2$ matrix and $\lambda_2 (\cdot)$ the smaller absolute value eigenvalue of a $2\times2$ matrix, i.e. $|\lambda_1 (\cdot)| ...
10
votes
4answers
839 views

Bounding the absolute sum of entries of the inverse of a 0-1 matrix

I have a non-singular square 0-1 matrix and I want to bound the sum of absolute values of its inverse as a function of n (or the vector 1-norm). Asymptotic results are also useful. Does anyone know ...
5
votes
1answer
1k views

Stochastic Matrix: Second largest eigenvalue and second largest absolute value of eigen value

Setup Let $A$ be a stochastic matrix. Let the eigenvalues of $A$ be $1 = \lambda_1 \geq \lambda_2 \geq \lambda_3 ... \geq -1$. Let $\lambda = \max_{x: x \perp 1} \frac{||Ax||}{|| x ||}$ Question: ...
4
votes
2answers
984 views

bounding the trace of a matrix product by the operator norms; generalized Hölder inequality?

Let $A_i$ with $i=1,\dots,N$ and $p$ be real $M\times M$ matrices. Further, let $p$ be positive definite, i.e., $p\succ 0$, with $Tr(p)=1$. Let $0< a_i<1$ and $\sum_{i=1}^N a_i = 1$. Claim: ...
2
votes
1answer
484 views

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 ...
4
votes
1answer
211 views

Extensions to the Golden-Thompson inequality?

Let $A$ and $B$ be two Hermitian matrices. The famous Golden-Thompson inequality states that $$\text{tr}(e^{A+B}) \le \text{tr}(e^Ae^B)$$ However, for determinants we have equality $$\det(e^{A+B}) ...
9
votes
2answers
1k views

Question on eigenvalue square root subadditivity

ORIGINAL QUESTION Let $\lambda_{1}\left(\cdot\right)$ be the larger eigenvalue of a $2\times2$ matrix and $\lambda_{2}\left(\cdot\right)$ the smaller eigenvalue of a $2\times2$ matrix. Is it true ...
3
votes
1answer
733 views

Cauchy-like inequality for Kronecker (tensor) product

General question first: upper/lower bound a sum of Kronecker products by its components. More specifically, how is $$ \Vert\sum_{\alpha}S_{\alpha}\otimes B_{\alpha}\Vert$$ bounded by the operator ...
4
votes
3answers
589 views

Is the componentwise square-root of a positive-definite matrix also pos.-def.?

Let $A=(a_{ij}) \in \mathbb{R}^{n \times n}$ be a matrix with $a_{ij} = a_{ji} \geq 0$ and $B=(b_{ij})$ with $b_{ij} = \sqrt{a_{ij}}$. Is $B$ positive-definite whenever $A$ is? In other words: ...
3
votes
2answers
713 views

Tight bound for sum of entries of the inverse of a nonnegative matrix

While playing around with certain non-negative matrices, I got stuck at the following question. Let $A$ be a strictly positive-definite $n \times n$ matrix ($n \ge 3$), with ones on the diagonal, and ...