All Questions
Tagged with matrices inequalities
124 questions
1
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Inequality between the singular values for a sum of two matrices
For two complex matrices $A,B \in \mathbb{C}^{n\times m}$ how to prove that:
\begin{equation}
\overline{\sigma}(B-A) \ge \underline{\sigma}(B) - \underline{\sigma}(A)
\end{equation}
where $\underline{...
1
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1
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358
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Matrix Submodular Inequality
Given $a,b,x > 0$ I know following the submodularity property holds:
\begin{align}
\frac{1}{a} - \frac{1}{a+x} \geq \frac{1}{a+b} - \frac{1}{a+b+x}
\end{align}
My question is, does this property ...
1
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1
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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
1
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1
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144
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Matrix inequalities for the moment of the fixed Shatten norm
Let $A_i, i=1, \ldots, N$ be real (or complex) matrices of the same dimension. Let $r_i, i=1, \ldots, N$ be independent Rademacher random variables.
The following inequality gives a bound on the ...
- 343
1
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1
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231
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Unitary matrix and matrix inequality [closed]
Dear all,
Suppose U and V are unitary matrix, A and B are positive definite,
Does:
$UAU^{-1} < VBV^{-1}$
implies $A< B$
and vice versa?
- 77
1
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1
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719
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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
...
- 13
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0
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216
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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
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0
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103
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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 $A_1,\dots,A_r\in M_{n}(\mathbb{C})$ ...
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0
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About means of positive definite matrices
This question is related to this recent one.
Recall that on the cone $SPD_n$ of $n\times n$ symmetric positive definite matrices, there are various notions of means, which extend the same notions ...
- 52.3k
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422
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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
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0
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111
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Matrix eigenvalues inequality (2)
Suppose 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 $m \times m$ positive matrix, whose eigenvalues are $b_1\ge b_2\ldots \ge b_m>0;$ ...
- 119
1
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1
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Is it true that $B \leq I \rightarrow B^2 \leq I$ (or higher powers) for positive semidefinite $B$, mimicking positive scalars? [closed]
If we know that $B \geq 0 $ (positive semidefinite) and that $I-B \geq 0$, is it necessarily true that $I-B^2 \geq 0$?
- 11
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0
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55
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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 $p > 1$
$$A \succeq B \succeq 0 \Rightarrow A^p \...
- 11
1
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0
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85
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What are good bounds on ratios of subdeterminants?
Let $A$ be a symmetric matrix and $A_i$ be the matrix obtained from $A$ by dropping the $i^{th}$ row and column. Then what are some good bounds on the value of $\frac{det(A_i)}{det(A)}$ ?
Using the ...
- 1,893
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0
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396
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Bound of spectral radius of polynomial of a complex matrix
I am trying to prove or disprove the following inequality.
$$ ||P(A)||_2\leq 2 \max_{\alpha\in W(A)}| P(\alpha)|,$$
where $P(\cdot)$ is a complex polynomial, $A\in \mathbb{C}^{n\times n}$ and $W(A)...
- 163
1
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0
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112
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Tools to bound the singular values of a finite sum of random matrices from below?
Matrix Chernoff bounds (see also this arXiv paper) are usually used to give upper bounds on the largest eigenvalue of a finite sum of random matrices. Sometimes it can also be used to give a lower ...
- 111
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1
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109
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Could someone help me to prove or disprove the following inequality?
Let $(c_{nr})$ be an $N\times R$ complex matrix, then $\forall z_n \in \mathbb{C}$, we have
$$
\sum_r \Big|\sum_n c_{nr}z_n\Big|^2 \geq \frac{1}{\sigma_{max}} \sum_n |z_n|^2
$$
where $\sigma_{max}$ is ...
- 395
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1
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81
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A matrix inequality involving a singular matrix
I have three matrices $A \in \mathbb R^{n \times n}$, $B \in \mathbb R^{n \times n}$, and $X \in \mathbb R^{n \times n}$.
Suppose that $A$ is singular, $B = B^\top > 0$ and $X = X^\top > 0$.
...
- 85
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2
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124
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Bounding $E[\|\Sigma^{-1/2}(X-\mu)\|_2^3]$ for 2-dimensional Bernoulli
Let $X\in\{0,1\}^2$ have mean $\mu=\left[\begin{smallmatrix}p_1\\p_2\end{smallmatrix}\right]$ and $\Pr[X_1 = X_2 = 1] = p\le \min\{p_1,p_2\}$.
(Note we must have $1-p_1-p_2+p\ge 0$ for the ...
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57
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Class of covariance matrices invariant under permutations
I am reading a paper on covariance matrix estimation, and in this paper is introduced a class of covariance matrices:
\begin{equation}
U(q, c_0(p),M)=\{\Sigma: \sigma_{ii}\leq M,\quad \max_j\sum_{j=1}^...
- 1
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68
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Inequality between product of companion matrices and power of Pisot number
Let $d\geqslant 2$ be an integer and consider a convergent sequence of "companion" matrices
$$A_k := \begin{pmatrix}
a_{k,1} & a_{k,2} & \cdots & a_{k,d} \\\
& ...
- 101
0
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0
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189
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The algebraic connectivity of $P_n$, the path on $n$ vertices does not exceed $\frac{12}{n^2-1}$
The algebraic connectivity of $P_n$, the path on $n$ vertices does not exceed $\frac{12}{n^2-1}$.
Let algebraic connectivity of $P_n$ be denoted by $\mu$. I have proved a result that if $G$ is a ...
- 199
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0
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89
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Show that a certain ratio of diagonal entries dominates a certain ratio of singular values
Let $D=[d_{ij}]_{i,j=1,\ldots,4}$ be a $4 \times 4$ “density matrix”, that is, a Hermitian (possibly symmetric) positive definite matrix having trace 1—that is, the (nonnegative) diagonal entries sum ...
- 723
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Holder inequality for a general rectangular matrix
Let $A \in \mathbb{R}^{m\times n}$ and $p,q \in \mathbb{R}^{+}$ such that $\frac{1}{p}+\frac{1}{q}=1$. I am interested to prove the following:
$$ \|A\|_{p}=\|A^T\|_q$$
I have tried using Holder ...
