All Questions
Tagged with matrices inequalities
124 questions
53
votes
7
answers
51k
views
Determinant of sum of positive definite matrices
Say $A$ and $B$ are symmetric, positive definite matrices. I've proved that
$$\det(A+B) \ge \det(A) + \det(B)$$
in the case that $A$ and $B$ are two dimensional. Is this true in general for $n$-...
- 541
35
votes
3
answers
4k
views
A curious determinantal inequality
In my study, I come across the following curious inequality, which I do not know a proof yet (so I am asking it here).
Let $A, B$ be $n\times n$ (Hermitian) positive definite matrices. It is very ...
- 1,748
26
votes
3
answers
17k
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 ...
- 837
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
21
votes
0
answers
868
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 ...
- 28.6k
19
votes
1
answer
856
views
A possible extension of a determinant inequality
It is well known that if $A, B$ are positive semidefinite matrices, then $$\det (A+B)\ge \det A+\det B.$$
I am considering a possible extension of this result. Let $\mathbb{M}_m(\mathbb{M}_n)$ ...
- 1,748
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
17
votes
1
answer
2k
views
Hlawka inequality for determinants of positive definite matrices
It is mentioned here that if $A, B, C\in M_{n}(\mathbb C)$ are positive semidefinite, then $$\det (A+B+C)+\det C\ge \det (A+C)+\det (B+C)$$ (quoted from this article) and the special case ($C=\bf 0$) $...
- 13.4k
16
votes
5
answers
3k
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 ...
- 295
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
$$N([A,B])\le C_N N(A)N(B),\qquad\forall A,B\in{\bf M}...
- 13.4k
16
votes
2
answers
536
views
What is $A+A^T$ when $A$ is row-stochastic ?
This is motivated by this MO question.
If $A\in{\bf M}_n({\mathbb R})$ is row-stochastic (entrywise non-negative, and $\sum_j a_{ij}=1$ for all $i$), then $M:=A+A^T$ is
symmetric,
entrywise non-...
- 52.3k
16
votes
0
answers
488
views
An inequality for matrix norms
Working on a problem in combinatorics I come up with the following inequality on matrix norms, which I checked it also numerically:
Let $A=(a_{ij})$ be a real symmetric $n\times n$ matrix with ...
- 4,484
16
votes
0
answers
808
views
Determinant inequality involving Hermitian, positive definite matrices
Let $A,B,C\in M_{n}(\mathbb C)$ be Hermitian and positive-definite matrices such that $A+B+C=I_{n}$.
Show that
$$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$
This question has been ...
- 269
15
votes
2
answers
477
views
matrix inequality with orthogonal matrices
I would like to know if for $A,B\in SO(3)$ the inequality
$$
\|AB-BA\|_F\leq \|A-I\|_F\|B-I\|_F
$$
holds, where $\|\cdot\|_F$ denotes the Frobenius norm and $I$ the identity matrix. Using the identity
...
- 883
14
votes
4
answers
1k
views
An inequality on some pairs of orthogonal vectors
Let $n,k\geq 1$. Suppose that
$a_1, \ldots, a_n\in \mathbb{R}^k$, $b_1, \ldots, b_n\in \mathbb{R}^k$
and $a_i^T b_i = 0$ for $i=1,\dots, n$. Is it true that
$$
\sum_{i=1}^n \|a_i\|_2^2 + \sum_{i=1}^n \...
- 1,991
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
13
votes
2
answers
1k
views
A log inequality for positive definite trace-one matrices
Let $\{v_i\}_{i=1}^N$ be a set of $n$-dimensional real vectors and let $X=X^\top\in\mathbb{R}^{n\times n}$ be a positive definite trace-one matrix. I would like to prove (or disprove) the following ...
- 2,712
13
votes
1
answer
1k
views
An inequality for the spectral radius of matrices used by J. Bochi
I am interested in the history of an inequality for the spectral radius of a $d\times d$ real or complex matrix, which occurs in Jairo Bochi's 2002 article Inequalities for numerical invariants of ...
- 6,206
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
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
12
votes
1
answer
3k
views
Exchange determinant and integral of a matrix-valued function
Assume $A(x)=(a_{ij}(x))_{k\times k}$ is a Hermitian matrix function on some manifold $M$, is there any inequality relates the integral of its determinant $\int_M \det(A)$ and the determinant of its ...
- 195
12
votes
0
answers
218
views
Which ordering of factors is needed to obtain this kind of determinantal inequalities?
Let $A$ and $B$ be $n\times n$ Hermitian positive definite matrices. The curious determinantal inequality given here, which can be stated as $$\det (A^{4}+ ABBA+BAAB+B^{4})\ge\det(A^{4}+ AABB+BBAA+B^{...
- 13.4k
11
votes
1
answer
1k
views
A square root inequality for symmetric matrices?
In this post all my matrices will be $\mathbb R^{N\times N}$ symmetric positive semi-definite (psd), but I am also interested in the Hermitian case. In particular the square root $A^{\frac 12}$ of a ...
- 5,380
10
votes
2
answers
7k
views
Bounding the trace of a matrix product by the operator norms; generalized Hölder inequality?
$\DeclareMathOperator\Tr{Tr}$Let $A_i$ with $i=1,\dotsc,N$ and $p$ be real $M\times M$ matrices.
Further, let $p$ be positive definite, i.e., $p\succ 0$, with $\Tr(p)=1$. Let $0< a_i<1$ and $\...
- 103
10
votes
1
answer
615
views
A curious determinantal inequality I
Let $A, B$ be Hermitian matrices. Does the following hold?
$$\det(A^{2}+B^{2}+|AB+BA|)\leq \det(A^{2}+B^{2}+|AB|+|BA|)$$
As usual, $|X|=(X^*X)^{1/2}$. Clearly, quantities on both sides are no less ...
- 1,748
10
votes
1
answer
3k
views
Reverse Minkowski (and related) Determinant Inequalities
For positive semidefinite matrices $A,B,C \in \mathbb{R}^{n\times n}$, the following inequalities are well known:
$$(\det(A+B))^{1/n} \geq (\det A)^{1/n} + (\det B)^{1/n} $$
and
$$\det(A+B+C) + \...
- 716
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
9
votes
2
answers
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 ...
- 91
9
votes
2
answers
1k
views
$2$-norm distance between square roots of matrices
Suppose two square real matrices $A$ and $B$ are close in the Schatten 1-norm, i.e. $\|A-B\|_1=\varepsilon$. Can this be used to put a bound on the Schatten 2-norm distance between their square roots. ...
- 191
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
9
votes
1
answer
700
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
$$\sum_{i,j} K_{ij} f_i f_j \le \sum_{ij} K^\prime_{ij} f_i ...
- 4,000
9
votes
1
answer
804
views
A singular value-eigenvalue inequality
Singular value or eigenvalue problems lie at the center of matrix analysis. One classical result is
$$\lambda_{j}(X^{*}X+Y^{*}Y)\geq 2\sigma_j(XY^*)$$
for $j \in \{1, \ldots, n\}$, where $\lambda_j(\...
- 1,748
8
votes
3
answers
595
views
Jensen-like inequality for random matrix: $\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$
Let $X\in M_n(\Bbb R)$ be a random matrix with iid elements following a continuous distribution.
What are the necessary and sufficient conditions for $$\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$$ to hold? Is ...
- 1,459
8
votes
1
answer
678
views
Inequality involving tensor product of orthonormal unit vectors
Let $e_1,...,e_r$ be the first $r$ standard basis of $\mathbb{R}^n, r<n$. Let $u_1,...,u_n$ be another orthonormal basis of $\mathbb{R}^n$. Let $\otimes$ be the tensor product on $\mathbb{R}^n$ and ...
- 1,495
8
votes
1
answer
412
views
Big triples in a matrix
Consider an $n\times n$ real matrix $A=(a_{ij})$ with non negative entries. Assume that
- the sum of the three largest entries in each row is a constant $R$ (the same for all rows),
- the sum of the ...
- 5,103
8
votes
1
answer
290
views
Distance from nonnegativity of some orthonormal vectors
Suppose that $1 < k < n$. Does there exist a constant $\beta > 0$, such that for every $k$ orthonormal vectors $f_1,\ldots,f_k \in \mathbb R^n$,
there exist $k$ orthonormal vectors with ...
- 1,991
8
votes
1
answer
726
views
A direct proof of a property of symmetric 2x2-determinants
Let $f(a,b,c)=\det\begin{pmatrix}a &b\\ b& c\end{pmatrix}\in\mathbb{R}[a,b,c]$ be the determinant of a $2 \times 2$ real symmetric matrix.
Let $f(x_i,y_i,z_i)\geq 0$, $x_i\geq 0$, $z_i\geq 0$ ...
- 14k
8
votes
1
answer
904
views
A generalized log inequality for positive definite trace-one matrices
Let $\{V_i\}_{i=1}^N$ be a set of $n\times m$, $n\geq m$, real matrices of full column rank and let $X=X^\top\in\mathbb{R}^{n\times n}$ be a positive definite trace-one matrix. Moreover, let $A^{1/2}=(...
- 2,712
8
votes
0
answers
400
views
When do we have $\|X - Y\| = \|\Sigma(X) - \Sigma(Y)\|$?
For any $X \in \mathbb{C}^{m\times n}$, let $\Sigma(X)$ be the "middle factor" in its SVD, so that $X = U\Sigma(X) V^H$ and the diagonal of $\Sigma(X)$ is arranged in descending order.
...
- 269
8
votes
0
answers
576
views
A rank inequality
Suppose
$$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
7
votes
4
answers
2k
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:
$\...
7
votes
1
answer
1k
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
$\text{trace}(X^{-1})\geq\text{trace}([\text{diag}(X)]^{-1})$,
...
- 43
7
votes
1
answer
1k
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 ...
- 52.3k
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
6
votes
3
answers
698
views
Is $1/\max(i,j)$ a bounded matrix on Hilbert spaces?
I would like to know if the infinite matrix $[\frac{1}{\max(i,j)}]_{i,j\geq 1}$ represents a bounded operator on $\ell^2(\mathbb{N}^\star)$.
It would be sufficient to know if the Lehmer matrix $[\frac{...
- 111
6
votes
2
answers
2k
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 ...
- 28.6k
6
votes
3
answers
1k
views
Norm of the upper triangular part of symmetric matrix
Let $D\in \mathbb{R}^{n\times n}$ denote a lower triangular matrix. With $\|\cdot\|$ denoting the spectral matrix norm, is there an estimate like
$$
\|D\| \leq C\|D+D^T\|,
$$
where $C>0$ is ...
- 261
6
votes
1
answer
3k
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:
...
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