All Questions
Tagged with matrices inequalities
124 questions
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
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
$$\sum_{i,j}(G^5)...
- 128k
5
votes
1
answer
241
views
Trace inequality under consideration of definiteness
Let $G \in \mathbb{R}^{3 \times 3}$ a symmetric, but indefinite matrix and $U \in \mathbb{R}^{3\times 3}$ a symmetric and positive definite matrix. I would like to prove the inequality
$$ \text{Tr} \...
- 51
5
votes
1
answer
197
views
The largest $\ell_p$-norm of a sum of rows of a Sylvester-Hadamard-Walsh matrix
Given any $n\in\mathbb N$, consider the the Sylvester-Hadamard-Walsh matrix $M=(a_{i,j})_{i,j\in 2^n}$ of size $2^n\times 2^n$ and for a number $p\in[1,\infty)$, let
$$\nu_{n,p}=\max_{F\subseteq 2^n}\...
- 41.8k
5
votes
1
answer
644
views
A conjecture about the submatrix of orthogonal matrix
Let $U$ be an $n\times n$ orthogonal matrix, i.e. $U\in\mathbb{R}^{n \times n}$. For any non-empty ordered sets $S_1,S_2\subset\{1,2,...,n\}$, define $U_{S_1S_2}$ to be an $|S_1|\times|S_2|$ submatrix ...
- 1,495
5
votes
0
answers
2k
views
A stronger Cauchy-Schwarz inequality for traces of compression matrices
Assume that $A$ and $B$ are contractions, so
$I-AA^T$ and $I-BB^T$ are positive-definite matrices. Let
$C=(I-AB^T)^{-1}(I-AA^{T})(I-BA^{T})^{-1}$, and show that:
$$Tr\left(\frac{1}{1-AA^T}\right)...
- 4,270
4
votes
1
answer
977
views
Ratio sum comparison on operators
It is known by the Lidskii inequality, that $\sum_{i=1}^n \left|s_i(S)-s_i(T)\right|\le\sum_{i=1}^n s_i(S+T)$,
where $s_i(S)$ is the $i$-th singular value of $S$.
How would one prove that
$$\sum_{i=1}^...
- 41
4
votes
1
answer
225
views
$\mathrm{diag}\left[(A+D)^{-1}\right] \ge \left[\mathrm{diag}(A)+D\right]^{-1}$?
Let $A\in\mathbb{R}^{n\times n}$ be a positive semidefinite matrix and $D\in\mathbb{R}^{n\times n}$ be a diagonal positive definite matrix. Let $\mathrm{diag}(X)\in\mathbb{R}^{n\times n}$ denote the ...
- 2,712
4
votes
1
answer
400
views
Extensions to the Golden-Thompson inequality?
Let $A$ and $B$ be two Hermitian matrices. The famous Golden-Thompson inequality states that
$$\text{tr}(e^{A+B}) \le \text{tr}(e^Ae^B)$$
However, for determinants we have equality
$$\det(e^{A+B}) ...
- 28.6k
4
votes
2
answers
222
views
Bound on sum of $n$th super-diagonal entries in a $2n$ by $2n$ PSD matrix
Let $A,B,C\in\mathbb{R}^{n\times n}$ be such that
$\left(\begin{array}{} A & B \\ B^T & C \end{array}\right)\succeq 0$. I would like to prove that
$$\mathrm{trace}\,B \le \sum_{i=1}^n \sqrt{\...
- 392
4
votes
1
answer
147
views
prove spectral equivalence bounds for inverse fractional power of matrices
The question is an extention to the answered question prove spectral equivalence bounds for fractional power of matrices.
Let $A, D \in \mathbb{R}^{n \times n}$ be two symmetric,positive definite and ...
- 75
4
votes
1
answer
278
views
Given a correlation matrix $B$. What correlation matrix A (maximizes / minimizes) the following: det(A+B)
Given correlation matrix $B$ (positive semi-definite with ones in the diagonal).
1)Find the correlation matrix $A$ which maximizes $\det\left(A+B\right)$.
2)Find the correlation matrix $A$ which ...
- 41
4
votes
1
answer
413
views
Lipschitz property of matrix function only depending on singular values
Let $f$ be a function from $\mathbb{R}^{n\times n}$ to $\mathbb{R}$ such that there exists another symmetric function $g$ (invariant under permutation of coordinates) from $\mathbb{R}^{n}$ to $\mathbb{...
- 1,495
4
votes
1
answer
3k
views
Cauchy-like inequality for Kronecker (tensor) product
General question first: upper/lower bound a sum of Kronecker products by its components. More specifically,
how is $$
\Vert\sum_{\alpha}S_{\alpha}\otimes B_{\alpha}\Vert$$
bounded by the operator ...
- 731
4
votes
1
answer
828
views
Determinant inequality of square-product sum of diagonal matrix and upper-triangular matrix
Recently, I have seen a matrix inequality but don't know how to prove it. The inequality goes as follows.
For an arbitrary $n\times n$ diagonal matrix $\mathbf{D}$ and an arbitrary upper-triangular ...
- 51
4
votes
1
answer
1k
views
Generalizing inequality relating Euclidean distance & Frobenius norm to Bregman divergences such as relative entropy & von Neumann divergence
Motivation- A Special Case
Supposing $A,B\in\mathbb{S}^{m\times m}$ are symmetric positive semi-definite (SPD) matrices and $\mathbf{x}\in\mathbb{R}^m$ is a unit vector where $\|\mathbf{x}\|=1$, we ...
- 607
4
votes
0
answers
455
views
Inequalities for trace/eigenvalues of product of multiple 2x2 matrices
Consider the matrix product $\prod_i^n A_i$,
where each $A_i$ is a $2\times2$ matrix having the form $A_i = \left( \begin{smallmatrix} \lambda + \alpha_i & -\beta_i \\ 1 & 0\end{smallmatrix}\...
- 695
4
votes
0
answers
676
views
Weyl-type inequality for non-Hermitian matrices?
What is the weakest known condition under which a Weyl-type eigenvalue perturbation inequality holds? Does some analogue hold for normal matrices, for example?
- 6,541
3
votes
2
answers
3k
views
Generalized Hölder's inequality for operator (subordinate) norms
While perusing the Matrix norms section of Wikipedia, I came across this generalized version of Holder's inequality.
$$
\|A\|_2^2 \leq \|A \|_1 \|A \|_\infty\,,
$$
where,
$$
\|A \|_p = \max_{\|x\|_p ...
- 35
3
votes
2
answers
171
views
a monotone relation for s-numbers
Assume $A, B$ are self-ajoint compact operators. Is it true that $\|A+iB\|\le \|2A+iB\|$? Do we have a stronger inequality $\prod_{k=1}^ns_k(A+iB)\le \prod_{k=1}^ns_k(2A+iB)$ or even stronger one $s_n(...
- 478
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
3
votes
1
answer
160
views
A matrix monotonicity question
Let $X\in\mathbb{R}^{n\times n}$ be a positive semi-definite matrix and $A\in\mathbb{R}^{n\times n}$ be a stable matrix, i.e. a matrix whose eigenvalues are strictly inside the left-half complex plane....
- 2,712
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
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
3
votes
2
answers
249
views
Extend an inequality on matrix norms
Let $A$ denote an $n \times n$ matrix, and $\sigma_i(\cdot)$ denote $i$-th largest singular value. Can we extend the following result to general $p \geq 1$?
For all $k = 1, \dots, n$,
$$ \sum_{i = 1}^...
- 153
3
votes
1
answer
80
views
prove spectral equivalence bounds for fractional power of matrices
Let $A, D \in \mathbb{R}^{n \times n}$ be two symmetric,positive definite and tri-diagonal matrices for that we know that they are spectrally equivalent, thus ist holds
$$ c^- x^\top D x \le x^\top A ...
- 75
3
votes
1
answer
297
views
Showing the positivity of the determinant of $\mathfrak{sp}(n)$ without making use of diagonalization
Let $\mathfrak{sp}(n)$ be the lie algebra of compact symplectic group $\mathrm{SP}(n)$, regarded as a compact form of $\mathfrak{sp}(2n,\mathbb{C})$, so we can talk about its (complex) determinant.
...
- 385
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 $a, b \in \mathbb R^d$,
$$|a - b|^p \leq 2^{p - 1} (|a|^{p - 2} + |b|^{p - 2}) |a - ...
- 708
3
votes
1
answer
427
views
Minimum upper bound for sum of the entries of the inverse covariance matrix
Let $x \in \mathbb{R}^n$ and $k$ is RBF kernel
$$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} = \...
3
votes
0
answers
49
views
Stability of matrix equation
Let $M=I+A\in \mathbb{R}^{n\times n}$ for a skew-symmetric matrix $A$ with $\|A\|<1$ in the spectral norm. Using the $LU$-decomposition of $M$, it is easy to construct a solution $L,U\in \mathbb{R}^...
- 435
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
2
votes
1
answer
1k
views
On an eigenvalue inequality
Let $\lambda_1 (\cdot)$ be the larger absolute value
eigenvalue of a $2\times2$ matrix and $\lambda_2 (\cdot)$
the smaller absolute value eigenvalue of a $2\times2$ matrix, i.e.
$|\lambda_1 (\cdot)| \...
- 29
2
votes
1
answer
942
views
A singular value inequality
Let $s_1,s_2: \mathbb{R}^{2\times 2} \mapsto \mathbb{R}_+$,
$s_{1}\left(\cdot\right)\ge s_{2}\left(\cdot\right)\ge 0$, be the
singular values of a $2\times2$ matrix. Is it true that
$$\left|s_{1}\...
- 173
2
votes
2
answers
264
views
Prove spectral equivalence of matrices
Let $A,D \in \mathbb{R}^{n\times n}$ be two positive definite matrices given by
$$
D =
\begin{bmatrix}
1 & -1 & 0 & 0 & \dots & 0\\
-1 & 2 & -1 & 0 & \dots & 0\\...
- 31
2
votes
1
answer
376
views
Matrix eigenvalues inequality (1)
Assume that $A$ is a $n\times n$ positive matrix, whose eigenvalues are $a_1\ge a_2\ldots \ge a_n>0;$ $B$ is a $p \times p$ positive matrix, whose eigenvalues are $b_1\ge b_2\ldots \ge b_p>0.$ ...
- 119
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
2
votes
1
answer
146
views
Finding matrices $A$ such that the entries of $A^n$ have specified signs
What techniques are there for ensuring nonnegativity of various entries of matrix powers?
Specific Question: Consider a matrix $A\in SL_2(\mathbb R)$. Let $(A^n)_{i,j}$ denote the $(i,j)$ entry of ...
- 1,329
2
votes
1
answer
337
views
Operator norm of triangular truncation on symmetric matrices
Inspired by this question.
It is known that for the matrix $T_n \in \mathcal{M}_n$ (the space of real-valued $n \times n$ matrices) defined by
\begin{equation*}
(T_n)_{ij} = \begin{cases}
1 & i \...
- 83
2
votes
1
answer
350
views
Inequality on diagonal entries of a matrix product
Let $A$ and $B$ be two Hermitian matrices and let $D$ be a diagonal matrix.
Does there exist any inequality involving the trace for the diagonal entries $(D A D A D A B)_{i,i}$?
I am looking for ...
- 21
2
votes
1
answer
1k
views
An inequality involving traces and matrix inversions
The following question kept me wondering for some time:
Given the symmetric matrices $A,B,C\in\mathbb{R}^{n×n}$ where $A$ and $C$ are positive definite (hence invertible), and $B$ is positive ...
- 53
2
votes
0
answers
100
views
An inequality related to Problem 10210 AMM 1992 No. 3
Problem. Let $A$ be a $N \times N$ real matrix whose $(i,j)$ entry is $a_{ij} \ge 0, \forall i, j$. Let $1$ denote $N\times 1$ all-ones vector. Prove that
$$N^2 1^\top A^\top A A^\top 1 \ge (1^\top A ...
- 1,053
2
votes
0
answers
160
views
An "almost" true inequality for Hermitian matrices
Let $A$ be an $N\times N$ Hermitian matrix. For $p+q$ even, consider the following inequality:
$$\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
51
views
What conclusions can I derive from this family of trace inequalities?
Problem. Let $n_1,\ldots,n_s,m_1,\ldots,m_s\ge 0$ be nonnegative integers and set $m := \sum_{i=1}^s m_i$ and $n := \sum_{i=1}^s n_i$. Let $\oplus$ be an operation on matrices which stacks them in a ...
- 283
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
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
$$\...
- 623
2
votes
0
answers
171
views
Bounding the distance between two matrix power sequences
Let $A,B$ be Hermitian matrices so that $0 \le A,B < I$ and also
$(1-\varepsilon)(I-B)\le I - A \le (1+\varepsilon)(I-B)$.
For every $t \in \mathbb{N}$, consider the matrix $A_{t} = \sum_{i=0}^{t}...
- 225
2
votes
0
answers
147
views
Is the following inequality true for the norm of Moore-Penrose pseudoinverses?
Let $L$ be a real, positive semi-definite, symmetric, square matrix, with pseudoinverse $L^{+}$. It can be shown for the operator norms $||.||_{op}$ that: if $L$ is invertible and $||I - L||_{op} < ...
- 1,465
1
vote
2
answers
86
views
Entrywise $\infty$-norm of squared difference of square roots of matrices
For a positive $n \times n$ definite real matrix $M$ we denote by $\sqrt{M}$ the positive square root of $M$. For an $n \times n$ matrix $A$ denote its entrywise infinity norm by
$$\|A\|_{\infty,\...
- 177
1
vote
1
answer
178
views
Matrix norm inequality for C*-Algebras [closed]
Let A a $C^*$-Algebra. I have already shown that the maps $Tr, \sigma: M_n(A)\rightarrow A$ given by $Tr((a_{ij})):=\sum_{i}a_{ii}$ and $\sigma\left(\left(a_{ij}\right)\right)=\sum_{i\text{, }j}a_{ij}$...
- 741