3
votes
1answer
40 views

Matrix inequalities question

Let $A, B \in \mathbb{R}^{n \times n}$. Assume that: $$ 0 \preccurlyeq 2 A^\top A \preccurlyeq A^\top + A $$ $$ B^\top + B \preccurlyeq 0 $$ Is the following inequality true? $$ A B + B^\top ...
1
vote
0answers
17 views

Bounding the norm of the product of random PSD matrices

Consider the following setup, $(X, \hat{X}, Y, \hat{Y})$ are four $n \times n$ real, symmetric, full-rank, positive-definite matrices with entries between zero and one and operator norm $O(n)$. The ...
0
votes
2answers
45 views

When does the equality hold for norm equivalence

We know that for a vector $x\in \mathbb{R}^n$, its 1-norm and 2-norm satisfy that $$\frac{1}{n}\|x\|_1\le\|x\|_2\le \|x\|_1,$$ could anyone please give me some hints that on what condition these ...
5
votes
0answers
132 views

Inequality involving traces and matrix inversions

The following question kept me wondering for some weeks: Given the symmetric matrices $A,B,C\in\mathbb{R}^{n\times n}$ where $A$ and $C$ are positive definite (hence invertible), and $B$ is positive ...
0
votes
0answers
19 views

$ || \lambda(A) - \lambda(B) ||_p \prec_k || \lambda(A -B) ||_p$?

Given two Hermitian matrices $\mathbf{A}$ and $\mathbf{B}$ and eigenvalue function $\lambda(\cdot)$ which returns eigenvalues of a matrix in non-increasing order. I found the following is true from ...
0
votes
0answers
61 views

equivalent LMIs

page 63 of LMI book of Stephen boyd: why and how LMI conditions 5.14 and 5.12 are equivalent? how to get from 5.12 to 5.14? [ A'P+PA+Lambda*C'*C PB+lambda*C'*D; (PB+lambda*C'D)' -lambda(I-D'*D)] ...
1
vote
1answer
46 views

Rank of a Matrix Sum

I have matrices of $3\times3$ dimension such that S=A+B. I know there is one inequality connecting rank of the matrices A,B and its sum S? Could you write down that here. It will be a great help for ...
2
votes
1answer
36 views

Equality case in the Frobenius rank inequality

In many linear algebra books, the following rank inequalities are found: Frobenius inequality Let $A$, $B$ and $C$ be three matrices such that the product $ABC$ is defined. Then ...
25
votes
0answers
524 views

How prove this matrix inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?

Question: let matrices $A,B,C\in M_{n}(C)$ be Hermitian and Positive definite matrices, such that:$$A+B+C=I_{n}$$ Show that: $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$ ...
0
votes
0answers
25 views

Finding matrix index from triangular array offset

I have a mapping from a lower triangular matrix, A, to a vector,v: A(i,j) -> v( $\lfloor i(i+1)/2 \rfloor + j$ ) $i,j\in[0,N]$, $j\leq i$, $N\in\cal{N}$, $N\geq 0$ (so, my first row is row 0, and ...
2
votes
1answer
69 views

Matrix Norm Inequalities and Linear Least Squares

I am working through one of my universities old QUALS and came across the following problem: Let $A,E\in \mathbb{C}^{m\times m}$. Suppose that $\sigma_\min>0$ is the smallest singular value of ...
2
votes
1answer
50 views

Can this multidimensional non-linear equation with constraints be minimized analytically?

I wish to find the vector of real numbers, $\mathbf{w}$, that minimizes the function: $$f(\mathbf{w}\mid\mathbf{p},\mathbf{q})=\sum_{t=0}^T \left[\left(\sum_{i=0}^I w_ip_{ti}\right)-q_t\right]^2,$$ ...
4
votes
3answers
80 views

Matrix inequality for square matrices

Does the following hold for any square matrix $A$, $(AA^*)^{1/2}\geq (A+A^*)/2$, where superscript $*$ denotes the Hermitian transpose. Proof/any comment would be appreciated.
0
votes
1answer
37 views

Matrix Inequality: $A^\top B A \preccurlyeq B$

Consider $A, B \in \mathbb{R}^{n \times n}$, $A$ invertible, $B \succ 0$. Say if the following holds: $$ A^{- \top} B A^{-1} \preccurlyeq B \ \Longleftrightarrow \ B \preccurlyeq A^\top B A. $$ I do ...
1
vote
1answer
48 views

Inequality of Weighted norm

I have a question about the weighted norm inequality: The weighted norm of a vector $x\in R^{M\times N}$ is defined by: $\left \| X\right \|_{w,*} = \sum_{_{i}}\left |w_{i}\sigma _{i}\left ( X ...
2
votes
1answer
59 views

Elementary matrix inequality

Let $A \in \mathbb{R}^{n \times n}$ be a positive semidefinite matrix. Is the mapping $$ \begin{align} F \ \colon \ \mathbb{R}^{n \times n} &\to \mathbb{R}^{n \times n} \\ X &\mapsto X^{-1} - ...
1
vote
0answers
38 views

Relation between determinant and L1 norm

Recently, I have coped with a problem about the relation between determinant of positive definite matrices and their L1 norm. More specifically, assume that $\Sigma_{1}$ and $\Sigma_{2}$ are two ...
4
votes
1answer
144 views

determinant inequality, $AB=BA$, then $ \det(A^2+B^2)\ge \det(2AB) $

$A$ and $B$ are two $n\times n $ real matrices, $AB=BA$. Can we conclude that $$ \det \Big(A^2+B^2\Big)\ge \det(2AB) $$ is right? Well, the inequality is interesting. if $A,B$ are upper ...
0
votes
1answer
23 views

Norm and Matrice Proof

I'm trying to show that the following statement is true: If $||\mathbf a - \theta \mathbf b||^2 - ||\mathbf a||^2 \geq 0$ for all $\theta \in [0,1]$, then $\mathbf a^T \mathbf b \le 0$. Is this ...
7
votes
1answer
247 views

A $2\times2$ Matrix inequality

$M,N$ are $2\times2$ real matrices, and $MN=NM$. Then, for any three real numbers $x,y,z$, we have $$4xz\det(xM^2+yMN+zN^2)\geq(4xz-y^2)\big(x\det(M)-z\det(N)\big)^2 $$ some thought: 1). ...
1
vote
1answer
57 views

Triangle inequality frobenius norm

I'm trying to show that the frobenius norm is a norm. however it appears as if triangle inequality isnt met. $$||A+B||_F = \sqrt{\sum_{i,j=1}^n |a_{ij}+b_{ij}|^2} \leq \sqrt{\sum_{i,j=1}^n ...
0
votes
0answers
41 views

Simple proof explanation - Possibly triangle inequality involved

I'd like some help with understanding the following statements...I saw it on the internet while searching for a proof, and I'd like to understand why its true: let $A$ be a diagonally dominant matrix ...
2
votes
2answers
48 views

2x2 Matrix Inequality

Is it true that if I have a positive definite matrix $m = \left( \begin{smallmatrix} m_{11} & m_{12}\\ m_{21} & m_{22} \\\end{smallmatrix} \right)$ in $\mathrm{M}(\mathbb{C};2)$ the following ...
3
votes
1answer
79 views

Inequality of Frobenius norm for skew matrices

Let $A$ be a complex skew-symmetric $n \times n$ matrix, that is, $A^T = -A$. Denote by $\|\cdot\|_F$ the Frobenius norm, that is, $\|B\|_F^2 = \text{trace}(B^*B)$. I would like to prove that $$ ...
0
votes
1answer
47 views

Prove relative error with condition number of matrix inequality

I was working on some questions and solutions, and encountered the following question. I am able to prove the inequality using the given information and some algebraic manipulation but the "within ...
1
vote
1answer
30 views

$A'B=I \rightarrow B'B \geq (A'A)^{-1}$

Could anyone please help me show that for general matrices it holds that $$A'B=I \rightarrow B'B \geq (A'A)^{-1}$$ Thanks.
1
vote
1answer
31 views

Show $||A||_2=(\sum\limits_{i,j=1…n} a^2_{ij})^{(1/2)}$ defines a Matrix Norm

Show $$\def\norm#1{\left\lVert{#1}\right\rVert_2{}}\norm A ={\left(\sum_{i,j=1\ldots n} a^2_{ij}\right)}^{1/2}$$ defines a Matrix Norm for $A\in\mathbb R^{n\times n}$ to show: $\norm{ AB}\le ...
1
vote
0answers
51 views

Eigenvalues of sum of two particular matrices

Let $A$ be a matrix with real eigenvalues, its maximum eigenvalue is $0$ and it has sum for rows equals to zero. Let $B$ be a matrix $\mathrm{diag}([1\,0\, ...\, 0])$ and let $I$ be the identity ...
0
votes
0answers
27 views

Eigenvalues inequalities. Sum matrices. I need an help asap. [duplicate]

Let $A$ whose sum for rows is 0, can I prove that the $ \lambda_i \left ( \begin{bmatrix} I & 0\\ 0 & 0 \end{bmatrix} +A\right ) $ are strictly less than $\lambda_i \left ( \begin{bmatrix} I ...
0
votes
2answers
64 views

Norm of a matrix and lower bound for its determinant

Assume that $M$ is a positive constant, $A=[a_{ij}]$ is a matrix, and $\vert a_{ij}\vert \geq M $ for all $1\leq i,j \leq n$. Also, assume that $\det(A) \neq 0$ .Can we conclude that there exists a ...
2
votes
2answers
68 views

How to prove this matrix bound

Let an $m$ by $n$ matrix $A\in\mathbb C^{m\times n}$. Denote $M=\max_i\sum_{j=1}^n|A_{ij}|$ and $N=\max_j\sum_{i=1}^m|A_{ij}|$. Prove for any two vectors $x\in\mathbb C^m$ and $y\in\mathbb C^n$, we ...
2
votes
2answers
116 views

Positive matrix and positive vector

Let $A \in \mathbb{R}^{n \times n}$ be a non-negative matrix, i.e. $A_{i,j} \geq 0$ $\forall (i,j)$. Let $x \in \mathbb{R}^n \setminus \{0\}$ be a non-negative vector, i.e. $x_i \geq 0$ $\forall i$. ...
2
votes
1answer
55 views

bound on trace-norm of product of matrices

Is it true that $$ \|ABA^\dagger\|_1\leq \|A\|^2\|B\|_1, $$ where $\|A\|_1$ is the trace norm, $\|A\|$ is the spectral norm, and $A$ and $B$ are square matrices?
1
vote
0answers
128 views

Cauchy Schwarz 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 $X$ , $Y \in \mathrm{R}^n$ are random vectors, is there a ...
5
votes
1answer
83 views

Square matrix and determinant inequality

Let $A, B, C$ be invertible $n \times n$ square matrices with $AC=CA$ and $B^2C^2=I_n$ Is $\det(ABC +CBA +A^2+I_n)$ always $\geq 0$?
1
vote
1answer
75 views

Solving least-squares with linear inequality constraints

Given: $$ \begin{aligned} A \in \mathbb{R}^{m \times n} \\ Y \in \mathbb{R}^{m \times 1} \\ i=1,2,...m:\quad\text{real numbers }M_i \gt m_i \ge 0 \\ Y_i \ge 0 \end{aligned} $$ I seek $X \in ...
0
votes
0answers
24 views

Proving inequality involving matrix variables

I have to prove that: $\tau=\sigma^2\left(p^2-p+\frac{\theta^2(p-1)^2}{p(\sigma^2+f'f)}\right)>0$ where $c-a'Aa \geq p^2 \sigma^2 + p\theta^2 -(\sigma^2 1_p+\theta ...
-1
votes
1answer
76 views

Eigenvalues of sum of Hermitian matrices

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4
votes
2answers
169 views

How prove this matrix inequality $\det(A+B)\ge 2^n\sqrt{\det(A)\det(B)}$

Question: Let $A_{n\times n},B_{n\times n}$ be positive Hermitian matrices. Show that $$\det(A+B)\ge 2^n\sqrt{\det(A)\det(B)}.$$ I know this $$\det(A+B)\ge \det(A)+\det(B)$$ But My problem I ...
2
votes
2answers
99 views

Find the Norm of Matrix using Cauchy-Schwarz inequality

Let $A$ be $n \times n$ matrix and such that all of its entries are uniformly $O(1)$. Using Cauchy-Schwarz inequality, show that the operator norm of matrix $A$, which is $\|A\|_{op}=\sup_{x\in R^n: ...
2
votes
1answer
75 views

How prove this matrix inequality $\lambda_{n-1}\le\frac{n}{n-1}\min{\{a_{jj}:1\le j\le n\}}$

let this Positive semi-definite matrix $A=(a_{ij})_{n\times n}$,and the Characteristic values is $\lambda_{1},\lambda_{2},\cdots,\lambda_{n}$,such $\lambda_{1}\ge \lambda_{2}\ge\cdots\ge ...
4
votes
1answer
165 views

Generalization of $\frac{a + b}{c + d} \leq \text{max}(\frac{a}{c}, \frac{b}{d})$

I'm looking for a matrix version of the basic inequality for the ratio of two sums of positive numbers: $$\frac{a + b}{c + d} \leq \max\left\{\frac{a}{c}, \frac{b}{d}\right\}.$$ Specifically, I have ...
1
vote
0answers
36 views

How show that $\det(a^{b_{i}}_{i}+1)+\det(a^{b_{i}}_{i}-1)>0$

Let $1\le a_{1}<a_{2}<\cdots<a_{n},1\le b_{1}<b_{2}<\cdots<b_{n}$, show that $$\begin{vmatrix} a^{b_{1}}_{1}+1&a^{b_{2}}_{1}+1&\cdots&a^{b_{n}}_{1}+1\\ ...
3
votes
3answers
245 views

How prove this matrix inequality $\det(B)>0$

Let $A=(a_{ij})_{n\times n}$ such $a_{ij}>0$ and $\det(A)>0$. Defining the matrix $B:=(a_{ij}^{\frac{1}{n}})$, show that $\det(B)>0?$. This problem is from my friend, and I have considered ...
1
vote
1answer
133 views

If $M$ is positive definite, then $\operatorname{det}{(M)}\leq \prod_i m_{ii}$

In the Wikipedia article on positive definite matrices they claim that if $M$ is positive definite, then the determinant of $M$ is bounded by the product of its diagonal entries. How might we show ...
0
votes
2answers
40 views

Matrix norm inequality involving max and stacked matrices

In a paper I found the following inequality for matrices $A$ and $B$: $\max\left\{||A||, ||B||\right\} \le \left\| \begin{align}A \\ B\end{align} \right\|_2 $ I suspect that this is a well-known ...
4
votes
0answers
125 views

Is $(tr(A))^n\geq n^n \det(A)$ for a symmetric positive definite matrix $A\in M_{n\times n} (\mathbb{R})$

If $A\in M_{n\times n} (\mathbb{R})$ a positive definite symmetric matrix, Question is to check if : $$(tr(A))^n\geq n^n \det(A)$$ What i have tried is : As $A\in M_{n\times n} (\mathbb{R})$ a ...
4
votes
1answer
73 views

Inequality with determinants problem

Let $A,B \in M_{2}(\mathbb{R})$ with $AB=BA.$ Prove that: $$\det(A^{2}+AB+B^{2})\geq (\det(A)-\det(B))^{2}$$
0
votes
1answer
180 views

Question about diagonal entries of inverse matrices?

Assume $A$ is a symmetric positive semidefinite matrix with diagonal zero and all other entries are less than one. Also assume $D$ is a diagonal matrix with all entries in diagonal are positive and ...
2
votes
0answers
19 views

Inequalities on Matrix Minimax?

Suppose I have a matrix $M$. How can I get a good bound on the minimax quantity $$ \min_{i}\max_{j}M_{ij} $$ or variations thereof? Links to literature would be greatly appreciated.