All Questions
Tagged with linear-algebra inequalities
176 questions
5
votes
0
answers
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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:
$$Tr\left(\frac{1}{1-AA^T}\right)...
CommunityBot
- 1
4
votes
1
answer
977
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Ratio sum comparison on operators
It is known by the Lidskii inequality, that $\sum_{i=1}^n \left|s_i(S)-s_i(T)\right|\le\sum_{i=1}^n s_i(S+T)$,
where $s_i(S)$ is the $i$-th singular value of $S$.
How would one prove that
$$\sum_{i=1}^...
CommunityBot
- 1
2
votes
1
answer
1k
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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 ...
- 1,748
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 $\mathbb{M}_m(\mathbb{M}_n)$ ...
- 28.6k
3
votes
1
answer
735
views
A similar Cauchy-Schwarz inequality with linear-algebra
Let $A$ be matrix in $M_{n}$ (i.e., $n\times n$ complex matrices), and $\|A\|\le 1$, we call it a contraction.
Assume that $A$ and $B$ are contractions such that
$I-AA^*$ and $I-BB^*$ are positive-...
- 28.6k
9
votes
3
answers
7k
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Set of Positive Definite matrices with determinant > 1 forms a convex set
While reading a paper An Arithmetic Proof of John’s Ellipsoid Theorem by Gruber and Schuster, I have a question on their proof.
Consider an $n\times n$ real symmetric and positive definite matrix $\...
CommunityBot
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1
vote
1
answer
1k
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Bounding the positive semi-definite matrix with its block diagonal matrix [closed]
Can we bound $\mathbf{A}$ with $\mathbf{A^*}$ as ${\bf{A}} \preceq {{\bf{A}}^*}$ where
\begin{equation}
{\bf{A}} = \left[ {\begin{array}{*{20}{c}}
{{{\bf{A}}_{11}}}&{...}&{{{\bf{A}}_{1N}}}\\
...
- 28.6k
8
votes
1
answer
2k
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A spectral inequality for positive-definite matrices
Question. Given a positive-definite $n \times n$ matrix $A = (a_{ij})$ with eigenvalues
$$
\lambda_1 \leq \cdots \leq \lambda_n ,
$$
is there a sharp upper bound for the product $\lambda_2 \cdots \...
CommunityBot
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1
vote
1
answer
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How to solve this optimization with the orthogonal constraint?
Problem
Supposing that $A$ is a symmetric real matrix and $\{\mathbf{w}\_i\}_{i=1}^n$ is any orthogonal basis on $\mathbb{R}^n$ such that $W^\top W=WW^\top=\mathbf{I}_n$ where $W=\left[\mathbf{w}_1\;\...
- 607
12
votes
2
answers
1k
views
Quadratic Farkas' Lemma?
The Farkas Lemma says that if a system of linear inequalities implies
yet another linear inequality, then this last inequality can be obtained by
taking a positive linear combination of the ...
1
vote
1
answer
177
views
inequality for a symmetric nonnegative matrix
Given $A$ symmetric and semidefinite positive, for each $x$
$$ x'Ax \geq \frac{1}{\Vert A\Vert} \Vert Ax \Vert^2 $$
This inequality appears at page 24 of "Introduction to Optimization" from Boris T. ...
- 25.8k
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
0
votes
1
answer
269
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Is there relationship between $f\left({\sum_i(\mathbf{v}_i^\top\mathbf{x})^2\lambda_i},\sum_j{(\mathbf{u}_j^\top\mathbf{x})^2\theta_j}\right)$ and $\sum_i\sum_j{(\mathbf{v}_i^\top\mathbf{u}_j)^2f(\lambda_i,\theta_j)}$ if $f$ is jointly convex?
Hello, everyone!
As we know that by Jensen's inequality, for jointly convex function $f$ and $\sum_ix_i^2=1$, we have
$$f(\sum_i{x_i^2\lambda_i},\sum_i{x_i^2\theta_i)}\leq\sum_i{x_i^2f(\lambda_i,\...
- 607
1
vote
0
answers
285
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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
5
votes
1
answer
4k
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Determinant of a sum of two matrices (one dominating the other)
Let $A$ and $B$ be two $n \times n$ real matrices such that:
$\forall i, j: a_{ij} \geq 0, b_{ij} \geq 0$
let $a_\max$ be the largest entry of $A$ and $b_\min$ be the smallest nonzero entry of $B$; ...
-1
votes
1
answer
809
views
On an eigenvalue inequality [closed]
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
1
vote
1
answer
720
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A question on gauge functions
In the second paragraph on Page 71 of the book Matrix Analysis by
Bhatia, 1997, it says ``as a consequence of (III.12) we have Theorem
III 4.4''. How can one get the inequality in Theorem III 4.4 from
...
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2
votes
1
answer
1k
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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)| \...
CommunityBot
- 1
10
votes
1
answer
2k
views
Multilinear generalization of Cauchy-Schwarz inequality
Let $V$ be a real vector space, and let $(\cdot,\cdot;\cdot,\cdot) : V^4 \to \mathbb{R}$ be a multilinear form with the following properties:
$(x,y;z,w) = (y,x;z,w) = (x,y;w,z)$ (symmetry in the ...
- 29.7k
1
vote
1
answer
2k
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Sufficient and necessary conditions on equality of Cauchy-Schwarz inequality of determinants
Cauchy-Schwarz inequality of determinants:
for $A_{n\times k}$, $B_{n\times k}$, and $B'B$ non-singular, we have
$|A'B|^2\leq |A'A||B'B|$
I was wondering what's the sufficient and necessary ...
- 11.6k
9
votes
2
answers
1k
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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 ...
- 9,486
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
3
votes
3
answers
1k
views
a "reverse Hadamard inequality"
Is there an inequality of the form $|\det(A)| \geq F(v_1, \ldots, v_n)$ for a real $n\times n$-matrix $A$ with columns $v_i$, $F \geq 0$?
- 52.3k
6
votes
2
answers
2k
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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 ...
- 52.3k
-2
votes
1
answer
470
views
Little conjecture about sums of reciprocals
Given a finite list $x_i$ of $N$ positive reals, it seems that $\sum_{i=1}^N x_i = \sum_{i=1}^N x_i {}^{-1} \Rightarrow \sum_{i=1}^N x_i \geq N$. Can anyone give me a proof?
- 39.9k
0
votes
2
answers
579
views
Linear algebra inequality
I'm wondering (hoping) if an inequality is true. Please can anyone help me?
Let $V$ be a complex vector space $dim_{\mathbb{C}}(V)=n$
with a hermitian scalar product $h$.
Let $v,a, b \in V$.
Is it ...
- 19.6k