All Questions
Tagged with matrix-theory inequalities
13 questions
0
votes
1
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69
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Inequality for extremal values of product of Hermitian matrices
I am looking for a reference to verify the following inequality, where $X$ and $Y$ are Hermitian positive semidefinite matrices:
$$
\lambda_n(X^{1/2}YX^{1/2}) = \lambda_n(XY) \leq \lambda_n(X)\...
5
votes
1
answer
510
views
A potential new norm for matrices and Horn's inequalities
I am investigating a function defined in terms of the singular values of matrices. Initially, I simplified the problem by focusing on the eigenvalues of $2 \times 2$ Hermitian, positive-definite ...
1
vote
0
answers
216
views
Schatten norm inequality
Let $A,B$ be two $n\times n$ matrices.
Find a lower bound of the $p$-th Schatten norm
$\|(A-B)(A-B)^\ast\|_{S_{p/2}}^{1/2}$ in terms of Schatten norm of $\|(AA^*+BB^*)\|_{S_q}$ for any relation ...
- 97
1
vote
0
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267
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Majorization for singular values of the difference of two matrices: $|\sigma(A)-\sigma(B)| \prec_w \sigma(A-B)$?
For two vectors $x$ and $y$ in $\mathbb{R}^n$, recall that $y$ weakly majorizes
$x$, denoted by $x\prec_w y$, if the sum of the $k$ largest entries of $x$ is smaller than or equal to that of $y$ for ...
- 269
1
vote
0
answers
373
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Upper and lower bounds on the entries of a matrix power
Say I have a non-negative square $n\times n$ irreducible stochastic matrix $A$ (i.e., each column sums to 1), for which the following holds:
$$A_{ij} > 0 \iff A_{ji} > 0.$$
I know that no more ...
- 111
3
votes
1
answer
421
views
Inequality for $AB + BA$ when $A,B\geq0$, reference request
Let $A,B\geq0$ be positive semidefinite matrices of arbitrary size $n\times n$. Denote by $\alpha$ and $\beta$ their largest eigenvalues.
It is well-known that the eigenvalues of the expression $AB +...
- 219
6
votes
1
answer
840
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Quantum inspired matrix inequality
While mimicking the union bound in quantum systems, we land on the following conjecture but don't know how to prove this. Given any complex-valued $n\times m$ matrix $A$. A sub-matrix of $A$ is ...
- 181
9
votes
1
answer
534
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Well known matrix inequality?
I suspect that the following matrix inequality is well known, but I can't find a reference or proof:
Given $n \times n$ symmetric matrices $A,B$ such that $I_n \leq A,B$, is the following true?
$${...
- 1,211
2
votes
1
answer
104
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Matrix inequality with arbitrary large ratios
Let $M = (m_{ij})$ be $n \times n$ symmetric positive definite matrix. Then it can be proven that
$$ M^{1/2}A M^{1/2} \succeq M^{1/2}D M^{1/2}\succ 0
$$
so
\begin{equation}
\lambda_{\min}(M^{1/2}A M^...
- 145
10
votes
1
answer
615
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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
2
votes
0
answers
171
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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
3
votes
2
answers
3k
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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
41
votes
4
answers
5k
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The sum of squared logarithms conjecture
I am searching for the first proof of (or counterexample to) the following conjecture.
(The sum of squared logarithms conjecture)
For all natural numbers $n$ and positive numbers $x_1,x_2, \ldots , ...
- 421