All Questions
74 questions
0
votes
0
answers
68
views
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
$$A_k := \begin{pmatrix}
a_{k,1} & a_{k,2} & \cdots & a_{k,d} \\\
& ...
- 101
3
votes
0
answers
118
views
A matrix-valued analogue of a classical inequality
Let $p \geq 4$ be an even integer. In the study of variational problems in $W^{1, p}$, it is handy to know that for $a, b \in \mathbb R^d$,
$$|a - b|^p \leq 2^{p - 1} (|a|^{p - 2} + |b|^{p - 2}) |a - ...
- 708
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:
$$\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} (A^q)_{ii} \geq \Big(\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} \Big) \Big(\...
- 593
5
votes
1
answer
474
views
An inequality for certain positive-semidefinite matrices
Suppose that $G=(G_{ij})$ is a positive-semidefinite symmetric matrix with the diagonal entries all equal $1$ and all off-diagonal entries $\le0$. Does it then necessarily follow that
$$\sum_{i,j}(G^5)...
- 128k
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 ...
- 5,103
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+...
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 $A_1,\dots,A_r\in M_{n}(\mathbb{C})$ ...
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 ...
- 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
$$ c^- x^\top D x \le x^\top A ...
- 75
3
votes
2
answers
840
views
An inequality for the spectral radius of block matrices
Let $d,m$ be positive integers. Suppose that $A_{i,j}$ is a $d\times d$-matrix with real entries whenever $i,j\in\{1,\dots m\}$.
Let $A$ be the $dm\times dm$ matrix that can be written as a block ...
3
votes
1
answer
427
views
Minimum upper bound for sum of the entries of the inverse covariance matrix
Let $x \in \mathbb{R}^n$ and $k$ is RBF kernel
$$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} = \...
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
$$ \text{Tr} \...
- 51
3
votes
0
answers
155
views
Frobenius inner product of a zero line-sum matrix and a doubly stochastic matrix
Let $A$, $B$ be two $n\times n$ real matrices.
Let $A$ be a zero line-sum matrix where each row sum and each column sum equals zero, i.e., $$\sum_{i=1}^{n}a_{ij}=\sum_{j=1}^{n}a_{ij}=0 $$ (it seems ...
- 71
11
votes
1
answer
1k
views
A square root inequality for symmetric matrices?
In this post all my matrices will be $\mathbb R^{N\times N}$ symmetric positive semi-definite (psd), but I am also interested in the Hermitian case. In particular the square root $A^{\frac 12}$ of a ...
- 5,380
6
votes
1
answer
446
views
Matrix inequality : trace of exponential of Hermitian matrix
I want to know whether the following inequality holds or not.
\begin{align}
(\mathrm{Tr}\exp[(A+B)/2])^2\leq(\mathrm{Tr}\exp A)(\mathrm{Tr}\exp B)\tag{1}
\end{align}
where $A, B$ are Hermitian ...
- 73
4
votes
0
answers
456
views
Inequalities for trace/eigenvalues of product of multiple 2x2 matrices
Consider the matrix product $\prod_i^n A_i$,
where each $A_i$ is a $2\times2$ matrix having the form $A_i = \left( \begin{smallmatrix} \lambda + \alpha_i & -\beta_i \\ 1 & 0\end{smallmatrix}\...
- 695
2
votes
0
answers
106
views
Connections between eigenvalues of $B$ and $A+iB$
Consider two symmetric and real matrices $A,B\in\mathbb{R}^n$ and definie $A+iB$. Note that $A+iB$ is not hermitian in this case. There are many results based on Brendixson and Courant-Fischer, saying,...
- 21
2
votes
0
answers
75
views
Case of equality in entrywise spectral radius bound
Let $A,B$ denote square matrices such that $\lvert A_{ij}\rvert\le B_{ij}$ for all $i,j$, and denote the spectral radius by $\rho$. From the Gelfand spectral radius formula it is easy to see that
$$\...
- 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 ...
- 4,484
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\...
- 1,495
-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 ...
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 ...
- 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 ...
- 1,495
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 ...
- 1,495
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{...
- 1,495
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{...
- 1,879
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 ...
- 4,484
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 $A,B$:
$$
\text{tr}(\exp{(A+B)}) \...
- 163
26
votes
2
answers
1k
views
Symmetric strengthening of the Cauchy-Schwarz inequality
In this great question by Nathaniel Johnston, and in its answers, we can learn the following remarkable inequality: For all $v,w \in \mathbb{R}^n$ we have
\begin{align*}
\|v^2\| \, \|w^2\| - \langle ...
- 12.5k
8
votes
1
answer
290
views
Distance from nonnegativity of some orthonormal vectors
Suppose that $1 < k < n$. Does there exist a constant $\beta > 0$, such that for every $k$ orthonormal vectors $f_1,\ldots,f_k \in \mathbb R^n$,
there exist $k$ orthonormal vectors with ...
- 1,991
8
votes
0
answers
576
views
A rank inequality
Suppose
$$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_{...
- 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?
$$\det(C+ xA) \det(C-xA) \le (\det C)^2$$
- 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{...
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(\...
- 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 ...
- 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 ...
- 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 $f(x_i,y_i,z_i)\geq 0$, $x_i\geq 0$, $z_i\geq 0$ ...
- 14k
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 $I-B^2 \geq 0$?
- 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 $...
user99354
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 ...
- 723
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
$$N([A,B])\le C_N N(A)N(B),\qquad\forall A,B\in{\bf M}...
- 13.4k
2
votes
0
answers
147
views
Is the following inequality true for the norm of Moore-Penrose pseudoinverses?
Let $L$ be a real, positive semi-definite, symmetric, square matrix, with pseudoinverse $L^{+}$. It can be shown for the operator norms $||.||_{op}$ that: if $L$ is invertible and $||I - L||_{op} < ...
- 1,465
10
votes
1
answer
3k
views
Reverse Minkowski (and related) Determinant Inequalities
For positive semidefinite matrices $A,B,C \in \mathbb{R}^{n\times n}$, the following inequalities are well known:
$$(\det(A+B))^{1/n} \geq (\det A)^{1/n} + (\det B)^{1/n} $$
and
$$\det(A+B+C) + \...
- 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 $p > 1$
$$A \succeq B \succeq 0 \Rightarrow A^p \...
- 11
2
votes
1
answer
380
views
Bounding entries of the inverse of certain zero-one matrices
It is known that the entries of the inverse of an (invertible) $n \times n$ zero-one matrix $A$ can grow exponentially in $n$. See this MO question:
Bounding the absolute sum of entries of the ...
- 1,081
18
votes
1
answer
1k
views
A curious eigenvalue inequality
Suppose $A, B$ are positive definite Hermitian matrices, $U$ is a unitary matrix such that $AUB$ is Hermitian. The spectral radius of a square matrix $X$ is denoted by $\rho(X)$. In my study, I ...
- 1,748
3
votes
1
answer
428
views
Inverse Hadamard determinant inequality
As far as I remembered there is an inverse Hadamard inequality for the determinant of the form
$$
|D|>\prod_j \sqrt{(a_{jj}^2-\sum_{i\neq j}a_{ij}^2)}
$$
providing all values in $(\cdot)>0$.
...
- 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
$\|A^{1-p}B^p\|\ge \|A\sharp_p B\|$, ...
- 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 $...
- 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?
- 6,541