Tagged Questions

3
votes
1answer
120 views

Norm inequality involving matrices

Let $A$ and $B$ be two definite positive symmetric $n \times n$ matrices. Prove or disprove that $$ \Vert AB - B^{-1} A^{-1} \Vert \geq \Vert AB - I \Vert $$ where $\Vert . \Vert$ is the Frobenius ...
5
votes
2answers
80 views

Determinant inequality about positive definite matrices.

Assume $A \in M_n(\Bbb{R})$ (not necessarily symmetric), and for $\forall \alpha\not=0$, $\alpha^TA\alpha>0$. Show that $$\det\left(\frac{A+A^T}{2}\right)\leq \det A.$$
4
votes
1answer
66 views

A matrix eigenvalue question

If $A, B, C$ are positive definite matrices of size $n$, is it true that $\lambda_j(A(B+C)^2A)\ge \lambda_j(AB^2A)$, $j=1, \dots, n$? $\lambda_j$ means the $j$-th largest eigenvalue.
1
vote
1answer
33 views

Lower bounds for inner product $x^\top y$

Cauchy-Schwartz provides an upper bound for the inner product $x^\top y$. Are there theorems that talk about lower bounds for this quantity? Assume $x\ge 0$ and $y\ge 0$ wlog.
2
votes
1answer
48 views

If $H$ has an $a\times b$ submatrix of all $1$s, please prove that $ab\le n$.

Let $H$ be an $n\times n$ matrix with entries $\pm1$. Its rows are mutually orthogonal. If $H$ has an $a\times b$ submatrix of all $1$s, please prove that $ab\le n$.
1
vote
2answers
54 views

How to show for a PSD matrix $A$ that $\left \| \left ( A+I \right )^{-1} \right \| \leq \frac{1}{1+\sigma _{\min}\left ( A \right )}$?

If $A \in \mathbb{C}^{n \times n}$ is positive semidefinite, show that $\left \| \left ( A+I \right )^{-1} \right \| \leq \frac{1}{1+\sigma _{\min}\left ( A \right )}$, where $\sigma _{\min}\left ( A ...
0
votes
0answers
48 views

Solving an overdetermined system of inequalities using null-space arguments

The solutions to a linear system of equations: $$A\cdot x = b$$ (where $x$ is a $(n\times 1)$ column vector, $b$ is a $(m\times 1)$ column vector and $A$ is $(m\times n)$ matrix) can all be ...
2
votes
1answer
75 views

For what A, If $A+A^T>0$ then $A^2+A^{2T}>0$?

let me know if I am wrong with the next with a real square matrix A. $A+A^T = \sqrt{A^2+A^{2T}+AA^T+A^TA} > 0$ This square root exists right? And because of this, the sum of all its elements are ...
7
votes
3answers
118 views

Prove the inequality: $\prod_{j=1}^ka_{jj}\leq\left(\frac{1}{k}\sum_{j=1}^k\lambda_j\right)^k.$

To all those who are eagerly awaiting a new question, all those who love math, I give this challenge and I hope for you good moments of reflection. Let $A=(a_{ij})_n$ a real nonnegative symmetric ...
2
votes
1answer
100 views

An inequality involving the norms of symmetric positive definite matrices

Given A and B two real symmetric positive definite matrices is it true that, for some norm $\|.\|$, this inequality holds $$ \|AB-I\| \leq \|A^2B^2-I\| \qquad ? $$
0
votes
1answer
94 views

Schur complement condition for positive definiteness still for complex matrices

Say that we're given the following matrices: $S\in M_{m,m}$ symmetric positive definite, $A\in M_{n,n}$, $X\in M_{n,m}$ and $Y\in M_{n,m}$. Actually I need to use a program that doesn't accept matrix ...
14
votes
1answer
247 views

Trace inequality for real matrices

Is there any general result characterizing real matrices $A$ such that $$[\mathrm{tr}(A)]^2\leq n\mathrm{tr}(A^2)?$$ I can see that the inequality holds if: all eigenvalues of $A$ are real (by the ...
3
votes
2answers
125 views

Positive definite matrix inequality

Can someone help me with the following proof involving positive definite matrices: Suppose $X\succ 0$ positive definite. Show that $X-v{v^T}\succ 0$ if and only if ${v^T}X^{-1}v \le 1$. Thanks in ...
2
votes
0answers
44 views

Symmetric matrices $M$ such that $x,y\geq 0$ implies $(x^t M y)^2\geq (x^t M x)(y^t M y)$

Is there a name for $n$-by-$n$ symmetric matrices $M$ such that for all $n$-dimensional non-negative-valued vectors $x,y$ we have $$(x^t M y)^2 \geq (x^t M x)(y^t M y)?$$ In particular I am ...
2
votes
1answer
53 views

To show the inequality $\|A\|\geq\max\{\|u_1\|,\ldots,\|u_q\|,\|\vec{v_1}\|,\ldots,\|\vec{v_q}\|\}$

Let $A\in$ $\mathbb{C}^{p\times q}$ with column $u_1,\ldots,u_q$ and rows $\vec{v_1},\ldots,\vec{v_p}$. show that $$\|A\|\geq\max\{\|u_1\|,\ldots,\|u_q\|,\|\vec{v_1}\|,\ldots,\|\vec{v_q}\|\}$$ and ...
0
votes
1answer
77 views

Inequality involving the trace of a matrix, and a matrix normalization

Let $U$ be a non-zero $(q\times q)$ matrix, and assume this matrix is normalized such that $tr(U^{\intercal}U)=1$. Let $R$ be a symmetric $(q\times q)$ matrix, and $N>0$ be a positive integer. ...
3
votes
3answers
132 views

Symmetric positive definite matrix inequality

Hi could you help me with the following: Show that for a symmetric positive definite matrix $B$, $$b_{ij} + b_{jk} + b_{ki} \leqslant b_{ii} + b_{jj} + b_{kk}$$ holds for any $1 \leqslant i,j,k ...
4
votes
3answers
216 views

prove that $\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n.$

If $A$ is a $m \times n$ matrix and $B$ a $n \times k$ matrix, prove that $$\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n.$$ Also show when equality occurs.
3
votes
1answer
192 views

Matrix norm inequality implying eigenvector norm inequality

For a matrix $A$ let $\|A\|$ be the norm given by $\|A\|=\sup_{v \neq 0}\frac{\|Av\|}{\|v\|}$ where $\|v\|$ is the Euclidian norm on the vector $v$. Suppose we have matrices $M$ and $S$ with leading ...
2
votes
3answers
100 views

Proving a matrix equality

I have 2 matrices: $A \in R^{nxn}$ is a non-singular matrix and $B \in R^{nxn}$ is a singular matrix. Here is the expression I need to prove: $$||A - B|| \ge ||A^{-1}||^{-1}$$ I dont understand why ...
0
votes
0answers
28 views

Solve inequality with the matrix [duplicate]

Possible Duplicate: p(n) is count of all n-digit numbers… Can anyone help me solve this inequality with the matrix? Task is to prove that this is true. n is any positive integer. $$\ 5 ...
3
votes
1answer
121 views

How to prove $\left\|\ln\left(e^{iH_1}e^{iH_2}\right)\right\|\leq\left\|H_1\right\|+\left\|H_2\right\|$?

Let $H_1$ and $H_2$ denote arbitrary Hermitian operators (finite dimensional) and let $\left\|\ldots\right\|$ denote the usual operator norm. I conjecture that $$ ...
3
votes
1answer
120 views

Dimension of solution space for system of linear inequalities

Let's say I have a system of inequalities: $Ax \leq g$ for some $A \in \mathbb{R}^{4\times4}$, $x \in \mathbb{R}^4$, $g \in \mathbb{R}^4$, and $A$ is full rank. Here, the $\leq$ denotes element-wise ...
4
votes
1answer
115 views

inequalty concering inverses of positive definite matrix

I don't find a way to prove this: given $A$, $B$, symmetric and positive definite: $$A>B \Rightarrow A^{-1} < B^{-1},$$ where $A>B$ means that $A-B$ is positive definite.
2
votes
2answers
376 views

Smallest eigenvalues of Sum of Two Positive Matrices

Let $C = A + B$, where $A$, $B$, and $C$ are positive definite matrices. In addition, $C$ is fixed. Let $\lambda (A)$, $\lambda (B)$, and $\lambda (C)$ be smallest eigenvalues of $A$, $B$, and $C$, ...
1
vote
2answers
263 views

An inequality on trace of product of two matrices

Suppose we have two positive semi-definite matrices of dimension n, $A$ and $B$ s.t. Tr$(A)$, Tr$(B)\le1$. Can we say anything about Tr$(AB)$? (Is Tr$(AB)\le1$ too?)
1
vote
1answer
458 views

Matrix Norm Inequality proof

I came across this inequality $$\|ABC\|_F \leq \|A\| \|B\|_F \|C\|$$ for all matrices $A$, $B$ and $C$, where $\|\cdot\|$ is the operator norm (max singular value) I do not know how to prove this, ...
1
vote
0answers
84 views

How to prove an inequality for a special structure of strictly triangular matrix

The problem I cause is attached below. I am trying to prove the inequality. By using small $M$, I found that the terms on the left side of the inequality are part of the terms of the expansion on ...
3
votes
1answer
225 views

Cauchy-Schwarz matrix inequality for random vectors

If $X$ and $Y$ are random scalars, then Cauchy-Schwarz says that $$| \mathrm{Cov}(X,Y) | \le \mathrm{Var}(X)^{1/2}\mathrm{Var}(Y)^{1/2}.$$ If $X$ and $Y$ are random vectors, is there a way to bound ...
4
votes
1answer
58 views

inequality on inner product

Let $x \in \Bbb R^n$ and $Q \in M_{n \times n}(\Bbb R)$, where $Q$ is hermitian and negative definite. Let $(\cdot,\cdot)$ be the usual euclidian inner product. I need to prove the following ...
1
vote
2answers
110 views

A hard proof of two matrix's elements

This is not duplicate of A matrix's element proof, but it is harder than that one. Given an constant $\alpha \in (0,1)$, and an $n \times n$ matrix $X$ whose all entries are between 0 and ...
0
votes
2answers
66 views

A matrix's element proof

Thanks again for copper.hat and Robert Israel's quick immediate reply. While I am modifying the questions, they've already given the answer. Now in this thread, I've changed it back to the original ...
1
vote
0answers
47 views

How to construct a matrix satisfying two semidefinite constraints

You are given matrices $A$, $B$ and $C$. $C$ is symmetric and positive semidefinite. How would you go about constructing a matrix $X \succeq 0$ such that $X \succeq AXA^T$ and $C \succeq BXB^T$? ...
1
vote
2answers
380 views

Inequality involving norm of matrix integral

This question seems basic but I could not find an answer. I have seen the inequality $$\left\|\int_a^b x(t) dt \right\| \leq \int_a^b \left\| x(t) \right\| dt $$ where $x(t) \in \mathbb{R}^n$ is a ...
1
vote
1answer
121 views

Proof on the inequality involving matrix splitting and trace operator

Suppose positive definite matrices $V, B, D\in\mathbb{R}^{n\times n}$ are given, where $D$ only contains diagonal entries of $V$, i.e., $D=diag(V)$, and $X, G\in\mathbb{R}^{n\times 2}$. Could the ...
5
votes
1answer
364 views

Least-squares left-inverse having smallest Frobenius norm

While trying to prove that the left-inverse of $A$ provided by the least-squares solution to $y=Ax$ has the smallest Frobenius norm, I am stuck at a point which I describe below: Let $B$ be any ...
0
votes
0answers
145 views

How to solve these inequations?

$C_i$ is a $k_i\times N$ matrix over finite field $\mathbb{F}_q$, where $i\in \{1,2,\ldots,K\}$, $k_i<N$, and $q<K$. My questions are 1) how to determine whether there is a $1\times N$ vector ...
4
votes
3answers
192 views

Does the Schur complement preserve the partial order

Let $\begin{bmatrix} A_{1} &B_1 \\ B_1' &C_1 \end{bmatrix}$, $\begin{bmatrix} A_2 &B_2 \\ B_2' &C_2 \end{bmatrix}$ be symmetric positive definite matrices and be ...