All Questions
Tagged with linear-algebra inequalities
52 questions with no upvoted or accepted answers
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
809
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
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 $$\det (A^{4}+ ABBA+BAAB+B^{4})\ge\det(A^{4}+ AABB+BBAA+B^{...
- 13.4k
8
votes
1
answer
531
views
How large can the dimension of a 'Span of powers of a finite field basis' be?
Let $p$ be a prime. For finite field $\mathbb{F}_{p^k}$ and $d\in\mathbb{Z}^+$, I am considering the following quantity, where we interpret the field $\mathbb{F}_{p^k}$ also as a $\mathbb{F}_p$-vector ...
- 89
8
votes
0
answers
413
views
Eigenvalues of cyclic stochastic matrices
Let's consider the following $n \times n$ cyclic stochastic matrix
$$ M= \begin{pmatrix}
0 & a_2 & & & &b_n \\\
b_1 & 0& a_3& &&& \\\
& b_2 & ...
- 181
8
votes
0
answers
577
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
$$M_{...
- 500
8
votes
0
answers
492
views
Strange determinant inequality $\det(C+ xA) \det(C-xA) \le (\det C)^2$
Let $A$ be an all-one $3$-by-$3$ matrix, let $C$ be a $3$-by-$3$ matrix, and let $x$ be a real number. How might one prove the following inequality?
$$\det(C+ xA) \det(C-xA) \le (\det C)^2$$
- 99
6
votes
0
answers
587
views
Lower bound on the sum of singular values for a sum of Hermitian matrices
Denote the eigenvalues of an $n\times n$ matrix $\mathbf{X}$ by $\lambda_i(\mathbf{X})$ and its singular values by $\sigma_i(\mathbf{X})$, $i=1,\ldots,n$. When $\mathbf{X}$ is Hermitian, we know that $...
- 907
5
votes
0
answers
169
views
Is there a sharper Golden–Thompson inequality?
For any two Hermitian matrices $A$ and $B$, the Golden–Thompson inequality
$$\mathrm{Tr} (e^A e^B) \geq \mathrm{Tr} \, e^{A + B}$$
holds, and it is known to be a strict inequality whenever $[A, B] \...
- 51
5
votes
0
answers
835
views
Gershgorin's 2nd theorem (disjoint circles): elementary proof?
Let $A \in \mathbb{C}^{n\times n}$ be a complex matrix. We let $a_{i,j}$ be the $\left(i,j\right)$-th entry of $A$ for all $i, j \in \left[n\right]$ (where $\left[n\right]$ denotes $\left\{1,2,\ldots,...
- 33.8k
5
votes
0
answers
586
views
A minimal eigenvalue inequality
Suppose $A$ is an $n\times n$ real symmetric positive definite matrix. Let $A^{(-1)}_{i,j}$ be the $n\times n$ matrix the entries $(i,i),\,(i,j),\,(j,i),\,(j,j)$ of which equal to the corresponding ...
- 2,239
5
votes
0
answers
2k
views
A stronger Cauchy-Schwarz inequality for traces of compression matrices
Assume that $A$ and $B$ are contractions, so
$I-AA^T$ and $I-BB^T$ are positive-definite matrices. Let
$C=(I-AB^T)^{-1}(I-AA^{T})(I-BA^{T})^{-1}$, and show that:
$$Tr\left(\frac{1}{1-AA^T}\right)...
- 4,280
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}\...
- 695
4
votes
0
answers
144
views
A Pythagorian inequality characterization of inner-product spaces
Let $(X,\|\cdot\|)$ be a real normed space. For any points $A$ and $B$ in $X$, let $AB:=\|A-B\|$. Suppose that for any points $A$ and $B$ in $X$ and any straight line $\ell\subseteq X$ such that $B\...
- 128k
4
votes
0
answers
166
views
Sum of eigenvalues is nonpositive
Let $A$ be a symmetric positive semidefinite $n \times n$ matrix. How can I show that the sum of the largest $n-k+1$ eigenvalues of $A - k\cdot \textrm{diag}(A)$ is nonpositive, for any $k \in \{1, \...
- 41
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
3
votes
0
answers
57
views
Maximizing a Gaussian quadratic form
Let $u$ denote a fixed unit vector in $\mathbb{R}^n$ and $g$ a standard Gaussian vector (in $\mathbb{R}^n$).
Consider the map
$$
f_n(X) = \mathbb{E} \langle (X^{-1} + gg^T)^{-1} u, u\rangle,
$$
...
- 390
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 $a, b \in \mathbb R^d$,
$$|a - b|^p \leq 2^{p - 1} (|a|^{p - 2} + |b|^{p - 2}) |a - ...
- 708
3
votes
0
answers
105
views
Techniques for solving linear inequalities
For $n$ real variables $x_1, \ldots, x_n$, I have a bunch of inequalities of form $2 x_i > x_j + x_k$ or $2 x_i < x_j + x_k$, where $i,j,k$ are distinct. My goal is to determine whether this set ...
- 231
3
votes
1
answer
428
views
Minimum upper bound for sum of the entries of the inverse covariance matrix
Let $x \in \mathbb{R}^n$ and $k$ is RBF kernel
$$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} = \...
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
3
votes
0
answers
77
views
A concentration problem of product of matrices
Let $A$ be an $n \times m$ matrix with non-negative entries and $B \in \mathbb{R^{n\times n}_{\geq 0}}, C \in \mathbb{R^{m\times m}_{\geq 0}}$ be random matrices where B and C are both symmetric and ...
- 195
3
votes
0
answers
158
views
Worst-Case Solution to (Stochastic) Matrix Inequality
EDIT: Some specific conjectures added.
This problem comes with an associated stochastic process, but I phrase everything as linear algebra in case somebody from a non-probability community has seen ...
- 31
3
votes
0
answers
385
views
Does this inequality of negative relative entropy and quantum relative entropy hold?
Hello, everyone!
Question
I have a question about the relationship between general relative entropy and general quantum relative entropy: Given a unit vector $|i\rangle$ and two Hermitian matrices $...
- 607
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:
$$\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} (A^q)_{ii} \geq \Big(\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} \Big) \Big(\...
- 593
2
votes
0
answers
80
views
Inequality involving minors of an orthogonal matrix
Fix $n \geq 3$ and take any orthonormal vectors $x,y,z \in \mathbb{R}^n$. Let also $A \in M_n(\mathbb{R})$ be a symmetric matrix with positive entries ($A_{ij} = A_{ji} > 0$). Is the following ...
- 21
2
votes
0
answers
97
views
Fractional reverse direction Cauchy-Schwarz inequality
If $Z_1,\dots,Z_r$ are complex $m\times m$-matrices, then let $\Phi(A_1,\dots,A_r):M_m(\mathbb{C})\rightarrow M_m(\mathbb{C})$ be the linear mapping defined by $\Phi(A_1,\dots,A_r)(X)=A_1XA_1^*+\dots+...
2
votes
0
answers
66
views
Minimizing a certain norm of the identity operator on $\mathbb R^2$
$\newcommand\R{\mathbb R}\newcommand\Q{\mathcal Q}$For mutually orthogonal vectors unit vectors $a=[a_1,\dots,a_n]^T$ and $b=[b_1,\dots,b_n]^T$ in $\R^n=\R^{n\times1}$ (so that $n\ge2$) and for all $x=...
- 128k
2
votes
0
answers
106
views
Connections between eigenvalues of $B$ and $A+iB$
Consider two symmetric and real matrices $A,B\in\mathbb{R}^n$ and definie $A+iB$. Note that $A+iB$ is not hermitian in this case. There are many results based on Brendixson and Courant-Fischer, saying,...
- 21
2
votes
0
answers
75
views
Case of equality in entrywise spectral radius bound
Let $A,B$ denote square matrices such that $\lvert A_{ij}\rvert\le B_{ij}$ for all $i,j$, and denote the spectral radius by $\rho$. From the Gelfand spectral radius formula it is easy to see that
$$\...
- 623
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} < ...
- 1,467
1
vote
0
answers
80
views
Inequality involving random vectors and absolute values
Let $\mathbb{X}, \mathbb{Y} \subset \mathbb{R}^d$ be finite sets. Suppose random vectors $X \in \mathbb{X}$ and $Y \in \mathbb{Y}$ are sampled according to a joint distribution $\mathbb{P}_{XY}$. ...
1
vote
0
answers
134
views
Are these $L_2$-spectral radii approximations strictly increasing whenever $(A_1,\dots,A_r)$ has no non-trivial irreducible subspace?
If $X$ is a complex matrix, then let $\overline{X}=(X^T)^*=(X^*)^T$ (this means that $\overline{X}$ is the matrix obtained by replacing every entry in $X$ with its complex conjugate), and let $\rho(X)$...
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 $A_1,\dots,A_r\in M_{n}(\mathbb{C})$ ...
1
vote
0
answers
65
views
A question about an inequality for hafnians of some special matrices
Let $S$ by a complex symmetric $2m$ by $2m$ matrix. Let $\sigma$ be the $2m$ by $2m$ matrix which is the direct sum of $m$ copies of the following matrix:
$$ \begin{pmatrix} 0 & 1 \\ 1 & 0 \...
- 5,215
1
vote
0
answers
63
views
The relationship of deteminant of a matrix and its diagonol block matrix?
I am trying to prove an inequality that :
$$
\operatorname{det} \Sigma_{V \mathrm{~V}}<\prod i \operatorname{det}\left(\Sigma_{V_{i} V_{i}}\right)
$$
Where
$$
\Sigma_{V V}=\left[\begin{array}{ccc}
\...
- 11
1
vote
0
answers
87
views
Approximation bounds for matrix multiplication
$\DeclareMathOperator{\op}{\mathrm{op}}$Since matrix multiplication is continuous, I expect that if $A_n\to A\in \mathrm{Mat}_{d\times d}(\mathbb{R})$ for the operator-norm and if $x_n\to x$ in $\...
- 5,405
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
vote
0
answers
110
views
Tail bound without independence
Suppose $X_i , X_j\in \mathbb{R}^d$ are gaussian vectors and $A$ is an $n\times n$ symmetric PSD matrix where $A_{ij} = f(\|X_i-X_j\|_2), \quad i,j\in 1,\ldots,n\;$ for some non-negative Lipschitz ...
- 195
1
vote
0
answers
282
views
Strict monotonicity of conditional variances
Let $K \geq 2$ be a positive integer and $C$ be any $K \times K$ non-singular matrix (if necessary, can assume that all $K$ rows of $C$ are needed to span the coordinate row vector $e_1'$). For ...
- 765
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 $p > 1$
$$A \succeq B \succeq 0 \Rightarrow A^p \...
- 11
1
vote
0
answers
85
views
What are good bounds on ratios of subdeterminants?
Let $A$ be a symmetric matrix and $A_i$ be the matrix obtained from $A$ by dropping the $i^{th}$ row and column. Then what are some good bounds on the value of $\frac{det(A_i)}{det(A)}$ ?
Using the ...
- 1,893
1
vote
0
answers
396
views
Bound of spectral radius of polynomial of a complex matrix
I am trying to prove or disprove the following inequality.
$$ ||P(A)||_2\leq 2 \max_{\alpha\in W(A)}| P(\alpha)|,$$
where $P(\cdot)$ is a complex polynomial, $A\in \mathbb{C}^{n\times n}$ and $W(A)...
- 163
1
vote
0
answers
112
views
Tools to bound the singular values of a finite sum of random matrices from below?
Matrix Chernoff bounds (see also this arXiv paper) are usually used to give upper bounds on the largest eigenvalue of a finite sum of random matrices. Sometimes it can also be used to give a lower ...
- 111
1
vote
0
answers
225
views
Is there an improvement for the Schur-Horn inequalities for positive semi-definite matrices?
By the Schur-Horn inequality I am thinking of the statement that for any Hermitian matrix $H$ its diagonal n-tuple $(H_{11},H_{22},..,H_{nn})$ for any choice of basis lies in the convex hull of the $n!...
- 1,893
1
vote
0
answers
285
views
Given a jointly convex function $f$, what is the bound of $f\left(\sum_ip_i^2x_i,\sum_jq_j^2y_j\right)$if $\mathbf{p},\mathbf{q}$ are constrained in a manifold?
Suppose there is a jointly convex function $f:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$, $\mathbf{x},\mathbf{y}\in\mathbb{R}^m$ and $\mathbf{p}=[p_1\;\ldots\;p_m]^\top,\mathbf{q}=[q_1\;\ldots\;...
- 607
0
votes
0
answers
68
views
Inequality between product of companion matrices and power of Pisot number
Let $d\geqslant 2$ be an integer and consider a convergent sequence of "companion" matrices
$$A_k := \begin{pmatrix}
a_{k,1} & a_{k,2} & \cdots & a_{k,d} \\\
& ...
- 101
0
votes
1
answer
158
views
Techniques for bounding the operator norm of the expectation of random matrix?
Let $\mu$ be a distribution on the unit sphere in $\mathbb{R}^n$. Let $u \sim \mu$ and consider the random matrix
$$
A = I_n - uu^T.
$$
Question: What techniques are available to provide (reasonably ...
- 390
0
votes
0
answers
177
views
Given optimality of L1 norm, prove that absolute value of sum of a vector with proper sign is less than 1?
Problem:
Given a domain $\mathcal{D}\subset\mathbb{R}^{l}$, we can find $l$
points $\boldsymbol{v}_{i}\in\mathcal{D}$, $i=1,\cdots,l$. Each
point is a column vector with dimension $l\times1$. They ...
- 1
0
votes
0
answers
69
views
Formulas involving traces of products of singular hermitian positive semidefinite $2$ by $2$ matrices
While working on the Atiyah problem on configurations of points, I came across formulas involving products of traces of products of singular hermitian positive semidefinite $2$ by $2$ matrices.
To ...
- 5,215