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Inequality between product of companion matrices and power of Pisot number

Let $d\geqslant 2$ be an integer and consider a convergent sequence of "companion" matrices $$A_k := \begin{pmatrix} a_{k,1} & a_{k,2} & \cdots & a_{k,d} \\\ & ...
Kermatoni's user avatar
  • 101
3 votes
0 answers
118 views

A matrix-valued analogue of a classical inequality

Let $p \geq 4$ be an even integer. In the study of variational problems in $W^{1, p}$, it is handy to know that for $a, b \in \mathbb R^d$, $$|a - b|^p \leq 2^{p - 1} (|a|^{p - 2} + |b|^{p - 2}) |a - ...
Aidan Backus's user avatar
2 votes
0 answers
160 views

An "almost" true inequality for Hermitian matrices

Let $A$ be an $N\times N$ Hermitian matrix. For $p+q$ even, consider the following inequality: $$\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} (A^q)_{ii} \geq \Big(\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} \Big) \Big(\...
WunderNatur's user avatar
5 votes
1 answer
474 views

An inequality for certain positive-semidefinite matrices

Suppose that $G=(G_{ij})$ is a positive-semidefinite symmetric matrix with the diagonal entries all equal $1$ and all off-diagonal entries $\le0$. Does it then necessarily follow that $$\sum_{i,j}(G^5)...
Iosif Pinelis's user avatar
8 votes
1 answer
412 views

Big triples in a matrix

Consider an $n\times n$ real matrix $A=(a_{ij})$ with non negative entries. Assume that - the sum of the three largest entries in each row is a constant $R$ (the same for all rows), - the sum of the ...
Yaakov Baruch's user avatar
2 votes
0 answers
97 views

Fractional reverse direction Cauchy-Schwarz inequality

If $Z_1,\dots,Z_r$ are complex $m\times m$-matrices, then let $\Phi(A_1,\dots,A_r):M_m(\mathbb{C})\rightarrow M_m(\mathbb{C})$ be the linear mapping defined by $\Phi(A_1,\dots,A_r)(X)=A_1XA_1^*+\dots+...
Joseph Van Name's user avatar
1 vote
0 answers
103 views

Is the rank preserved when the spectral radius is maximized?

If $A$ is a matrix, then let $\rho(A)$ denote the spectral radius of $A$. If $A=(a_{i,j})_{i,j}$, then let $\overline{A}=(\overline{a_{i,j}})_{i,j}$. Suppose that $A_1,\dots,A_r\in M_{n}(\mathbb{C})$ ...
Joseph Van Name's user avatar
4 votes
1 answer
147 views

prove spectral equivalence bounds for inverse fractional power of matrices

The question is an extention to the answered question prove spectral equivalence bounds for fractional power of matrices. Let $A, D \in \mathbb{R}^{n \times n}$ be two symmetric,positive definite and ...
Luna947's user avatar
  • 75
3 votes
1 answer
80 views

prove spectral equivalence bounds for fractional power of matrices

Let $A, D \in \mathbb{R}^{n \times n}$ be two symmetric,positive definite and tri-diagonal matrices for that we know that they are spectrally equivalent, thus ist holds $$ c^- x^\top D x \le x^\top A ...
Luna947's user avatar
  • 75
3 votes
2 answers
840 views

An inequality for the spectral radius of block matrices

Let $d,m$ be positive integers. Suppose that $A_{i,j}$ is a $d\times d$-matrix with real entries whenever $i,j\in\{1,\dots m\}$. Let $A$ be the $dm\times dm$ matrix that can be written as a block ...
Joseph Van Name's user avatar
3 votes
1 answer
427 views

Minimum upper bound for sum of the entries of the inverse covariance matrix

Let $x \in \mathbb{R}^n$ and $k$ is RBF kernel $$k(x, x') := \exp \left(-\frac{\|x-x'\|^2}{2\sigma^2}\right)$$ and let $\mathbf{K}$ be the following $n \times n$ covariance matrix $$\mathbf{K} = \...
Maryam Bahrami's user avatar
5 votes
1 answer
241 views

Trace inequality under consideration of definiteness

Let $G \in \mathbb{R}^{3 \times 3}$ a symmetric, but indefinite matrix and $U \in \mathbb{R}^{3\times 3}$ a symmetric and positive definite matrix. I would like to prove the inequality $$ \text{Tr} \...
TARS's user avatar
  • 51
3 votes
0 answers
155 views

Frobenius inner product of a zero line-sum matrix and a doubly stochastic matrix

Let $A$, $B$ be two $n\times n$ real matrices. Let $A$ be a zero line-sum matrix where each row sum and each column sum equals zero, i.e., $$\sum_{i=1}^{n}a_{ij}=\sum_{j=1}^{n}a_{ij}=0 $$ (it seems ...
Lo Celso's user avatar
11 votes
1 answer
1k views

A square root inequality for symmetric matrices?

In this post all my matrices will be $\mathbb R^{N\times N}$ symmetric positive semi-definite (psd), but I am also interested in the Hermitian case. In particular the square root $A^{\frac 12}$ of a ...
leo monsaingeon's user avatar
6 votes
1 answer
446 views

Matrix inequality : trace of exponential of Hermitian matrix

I want to know whether the following inequality holds or not. \begin{align} (\mathrm{Tr}\exp[(A+B)/2])^2\leq(\mathrm{Tr}\exp A)(\mathrm{Tr}\exp B)\tag{1} \end{align} where $A, B$ are Hermitian ...
s hukahi's user avatar
4 votes
0 answers
457 views

Inequalities for trace/eigenvalues of product of multiple 2x2 matrices

Consider the matrix product $\prod_i^n A_i$, where each $A_i$ is a $2\times2$ matrix having the form $A_i = \left( \begin{smallmatrix} \lambda + \alpha_i & -\beta_i \\ 1 & 0\end{smallmatrix}\...
Artemy's user avatar
  • 695
2 votes
0 answers
106 views

Connections between eigenvalues of $B$ and $A+iB$

Consider two symmetric and real matrices $A,B\in\mathbb{R}^n$ and definie $A+iB$. Note that $A+iB$ is not hermitian in this case. There are many results based on Brendixson and Courant-Fischer, saying,...
Stephan's user avatar
  • 21
2 votes
0 answers
75 views

Case of equality in entrywise spectral radius bound

Let $A,B$ denote square matrices such that $\lvert A_{ij}\rvert\le B_{ij}$ for all $i,j$, and denote the spectral radius by $\rho$. From the Gelfand spectral radius formula it is easy to see that $$\...
Julian's user avatar
  • 623
10 votes
1 answer
629 views

Minimum distance of a symmetric matrix to diagonal matrices

Let $A=(a_{ij})$ be an arbitrary $n \times n$ real symmetric matrix and $n\geq 2$. Let $\| \cdot \|$ denote the operator $2$-norm or equivalently the maximum absolute value of eigenvalues for ...
Mostafa - Free Palestine's user avatar
1 vote
0 answers
422 views

Integral of matrix determinant with respect to Lebesgue measure

$\newcommand\norm[1]{\lVert#1\rVert} \newcommand\opnorm[1]{\norm{#1}_{\text{op}}} \newcommand\Frnorm[1]{\norm{#1}_{\text F}}$Define \begin{align*} S_t=\{ (A,B)\in\mathbb{R}^{n\times n}\times\...
neverevernever's user avatar
-1 votes
1 answer
330 views

Holder inequality for a general rectangular matrix

Let $A \in \mathbb{R}^{m\times n}$ and $p,q \in \mathbb{R}^{+}$ such that $\frac{1}{p}+\frac{1}{q}=1$. I am interested to prove the following: $$ \|A\|_{p}=\|A^T\|_q$$ I have tried using Holder ...
Prashant Govindarajan's user avatar
3 votes
2 answers
375 views

Inequality for 0-1 matrices

Given an $n \times n$-matrix $A$ with entries only $0$ or $1$ and determinant equal to $\pm 1$. Define the magnitude $M_A$ of $A$ as the sum of all entries of the inverse of $A$. Question 0: Do we ...
Mare's user avatar
  • 26.5k
8 votes
1 answer
678 views

Inequality involving tensor product of orthonormal unit vectors

Let $e_1,...,e_r$ be the first $r$ standard basis of $\mathbb{R}^n, r<n$. Let $u_1,...,u_n$ be another orthonormal basis of $\mathbb{R}^n$. Let $\otimes$ be the tensor product on $\mathbb{R}^n$ and ...
neverevernever's user avatar
5 votes
1 answer
644 views

A conjecture about the submatrix of orthogonal matrix

Let $U$ be an $n\times n$ orthogonal matrix, i.e. $U\in\mathbb{R}^{n \times n}$. For any non-empty ordered sets $S_1,S_2\subset\{1,2,...,n\}$, define $U_{S_1S_2}$ to be an $|S_1|\times|S_2|$ submatrix ...
neverevernever's user avatar
4 votes
1 answer
413 views

Lipschitz property of matrix function only depending on singular values

Let $f$ be a function from $\mathbb{R}^{n\times n}$ to $\mathbb{R}$ such that there exists another symmetric function $g$ (invariant under permutation of coordinates) from $\mathbb{R}^{n}$ to $\mathbb{...
neverevernever's user avatar
9 votes
2 answers
912 views

A Matrix Inequality for positive definite matrices

Let $X$ and $Y$ be positive semi-definite self-adjoint complex matrices of same finite order. The, is it true that $|X-Y|\leq X+Y$ where for any matrix $A$, $|A|$ is defined to be $|A|:=(A^*A)^{\frac{...
A beginner mathmatician's user avatar
16 votes
0 answers
488 views

An inequality for matrix norms

Working on a problem in combinatorics I come up with the following inequality on matrix norms, which I checked it also numerically: Let $A=(a_{ij})$ be a real symmetric $n\times n$ matrix with ...
Mostafa - Free Palestine's user avatar
6 votes
1 answer
487 views

Intuitive proof of Golden-Thompson inequality

Sutter et al. [1] in their paper "Multivariate Trace Inequalities" give an intuitive proof of the following Golden-Thompson inequality: For any hermitian matrices $A,B$: $$ \text{tr}(\exp{(A+B)}) \...
Saket Choudhary's user avatar
26 votes
2 answers
1k views

Symmetric strengthening of the Cauchy-Schwarz inequality

In this great question by Nathaniel Johnston, and in its answers, we can learn the following remarkable inequality: For all $v,w \in \mathbb{R}^n$ we have \begin{align*} \|v^2\| \, \|w^2\| - \langle ...
Jochen Glueck's user avatar
8 votes
1 answer
290 views

Distance from nonnegativity of some orthonormal vectors

Suppose that $1 < k < n$. Does there exist a constant $\beta > 0$, such that for every $k$ orthonormal vectors $f_1,\ldots,f_k \in \mathbb R^n$, there exist $k$ orthonormal vectors with ...
Mahdi - Free Palestine's user avatar
8 votes
0 answers
576 views

A rank inequality

Suppose $$M := \begin{bmatrix} M_{11} & \cdots &M_{1d} \\ \vdots & \ddots & \vdots \\ M_{d1} & \cdots & M_{dd} \end{bmatrix}$$ is a $d \times d$ block matrix such that $$M_{...
SMD's user avatar
  • 500
8 votes
0 answers
491 views

Strange determinant inequality $\det(C+ xA) \det(C-xA) \le (\det C)^2$

Let $A$ be an all-one $3$-by-$3$ matrix, let $C$ be a $3$-by-$3$ matrix, and let $x$ be a real number. How might one prove the following inequality? $$\det(C+ xA) \det(C-xA) \le (\det C)^2$$
Martin's user avatar
  • 99
1 vote
1 answer
1k views

Inequality between the singular values for a sum of two matrices

For two complex matrices $A,B \in \mathbb{C}^{n\times m}$ how to prove that: \begin{equation} \overline{\sigma}(B-A) \ge \underline{\sigma}(B) - \underline{\sigma}(A) \end{equation} where $\underline{...
Valentin Preda's user avatar
9 votes
1 answer
804 views

A singular value-eigenvalue inequality

Singular value or eigenvalue problems lie at the center of matrix analysis. One classical result is $$\lambda_{j}(X^{*}X+Y^{*}Y)\geq 2\sigma_j(XY^*)$$ for $j \in \{1, \ldots, n\}$, where $\lambda_j(\...
M. Lin's user avatar
  • 1,748
10 votes
1 answer
615 views

A curious determinantal inequality I

Let $A, B$ be Hermitian matrices. Does the following hold? $$\det(A^{2}+B^{2}+|AB+BA|)\leq \det(A^{2}+B^{2}+|AB|+|BA|)$$ As usual, $|X|=(X^*X)^{1/2}$. Clearly, quantities on both sides are no less ...
M. Lin's user avatar
  • 1,748
14 votes
2 answers
574 views

A simple but curious determinantal inequality

Let $A$ and $B$ be $n\times n$ Hermitian positive definite matrices and $k>0$ real. Then $A^k$ is well-defined and experimentally, we have $$\det(A^k+BABA^{-1})\geqslant \det(A^k+BA^{-1}BA),$$or ...
Wolfgang's user avatar
  • 13.4k
8 votes
1 answer
726 views

A direct proof of a property of symmetric 2x2-determinants

Let $f(a,b,c)=\det\begin{pmatrix}a &b\\ b& c\end{pmatrix}\in\mathbb{R}[a,b,c]$ be the determinant of a $2 \times 2$ real symmetric matrix. Let $f(x_i,y_i,z_i)\geq 0$, $x_i\geq 0$, $z_i\geq 0$ ...
Dima Pasechnik's user avatar
1 vote
1 answer
95 views

Is it true that $B \leq I \rightarrow B^2 \leq I$ (or higher powers) for positive semidefinite $B$, mimicking positive scalars? [closed]

If we know that $B \geq 0 $ (positive semidefinite) and that $I-B \geq 0$, is it necessarily true that $I-B^2 \geq 0$?
bmath's user avatar
  • 11
12 votes
2 answers
4k views

Prove that matrix is positive definite

I faced a hard question in kernel methods theory, which I can't answer for about one week. Initially it was formulated in terms of positive valued functions, but it could be reformulated easier: Let $...
user avatar
0 votes
0 answers
89 views

Show that a certain ratio of diagonal entries dominates a certain ratio of singular values

Let $D=[d_{ij}]_{i,j=1,\ldots,4}$ be a $4 \times 4$ “density matrix”, that is, a Hermitian (possibly symmetric) positive definite matrix having trace 1—that is, the (nonnegative) diagonal entries sum ...
Paul B. Slater's user avatar
16 votes
2 answers
2k views

Bounding the matrix norm of a commutator $[A,B]$ in terms of the norms of $A$ and $B$

The setup is as in this question: Given a norm $N$ over ${\bf M}_n(\mathbb C)$, it is a natural question to find the best constant $C_N$ such that $$N([A,B])\le C_N N(A)N(B),\qquad\forall A,B\in{\bf M}...
Wolfgang's user avatar
  • 13.4k
2 votes
0 answers
147 views

Is the following inequality true for the norm of Moore-Penrose pseudoinverses?

Let $L$ be a real, positive semi-definite, symmetric, square matrix, with pseudoinverse $L^{+}$. It can be shown for the operator norms $||.||_{op}$ that: if $L$ is invertible and $||I - L||_{op} < ...
Learning math's user avatar
10 votes
1 answer
3k views

Reverse Minkowski (and related) Determinant Inequalities

For positive semidefinite matrices $A,B,C \in \mathbb{R}^{n\times n}$, the following inequalities are well known: $$(\det(A+B))^{1/n} \geq (\det A)^{1/n} + (\det B)^{1/n} $$ and $$\det(A+B+C) + \...
Tom's user avatar
  • 716
1 vote
0 answers
55 views

On the Lowener-Heinz inequality

I know that for two symmetric positive semi-definite (non-diagonal) matrices $A,B$, the inequality asserts that the following does not hold for all $p > 1$ $$A \succeq B \succeq 0 \Rightarrow A^p \...
user97064's user avatar
2 votes
1 answer
380 views

Bounding entries of the inverse of certain zero-one matrices

It is known that the entries of the inverse of an (invertible) $n \times n$ zero-one matrix $A$ can grow exponentially in $n$. See this MO question: Bounding the absolute sum of entries of the ...
Peter Dukes's user avatar
  • 1,081
18 votes
1 answer
1k views

A curious eigenvalue inequality

Suppose $A, B$ are positive definite Hermitian matrices, $U$ is a unitary matrix such that $AUB$ is Hermitian. The spectral radius of a square matrix $X$ is denoted by $\rho(X)$. In my study, I ...
M. Lin's user avatar
  • 1,748
3 votes
1 answer
428 views

Inverse Hadamard determinant inequality

As far as I remembered there is an inverse Hadamard inequality for the determinant of the form $$ |D|>\prod_j \sqrt{(a_{jj}^2-\sum_{i\neq j}a_{ij}^2)} $$ providing all values in $(\cdot)>0$. ...
Sergei's user avatar
  • 1,550
13 votes
2 answers
1k views

A matrix norm inequality

Suppose that $A, B$ are Hermitian positive definite matrices of the same order and $0\le p\le 1$. Using a standard approach in matrix analysis, one can show that $\|A^{1-p}B^p\|\ge \|A\sharp_p B\|$, ...
M. Lin's user avatar
  • 1,748
6 votes
0 answers
587 views

Lower bound on the sum of singular values for a sum of Hermitian matrices

Denote the eigenvalues of an $n\times n$ matrix $\mathbf{X}$ by $\lambda_i(\mathbf{X})$ and its singular values by $\sigma_i(\mathbf{X})$, $i=1,\ldots,n$. When $\mathbf{X}$ is Hermitian, we know that $...
Bullmoose's user avatar
  • 907
4 votes
0 answers
676 views

Weyl-type inequality for non-Hermitian matrices?

What is the weakest known condition under which a Weyl-type eigenvalue perturbation inequality holds? Does some analogue hold for normal matrices, for example?
Aryeh Kontorovich's user avatar