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Results tagged with matrix-analysis
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user 4312
The study of the properties of real and complex matrices that are more close to analysis and operator theory. For instance: the properties of positive definite matrices, matrix inequalities, perturbation analysis, matrix functions, inequalities between eigenvectors and singular values, majorization.
9
votes
Accepted
$\mathrm{diag}\left[(A+D)^{-1}\right] \ge \left[\mathrm{diag}(A)+D\right]^{-1}$?
This is true. As Darij suggests, denote $B=A+D$, let $e=e_i$ be a basic unit vector. We have to prove $(B^{-1}e,e)\geqslant (Be,e)^{-1}$. Denote further $C=B^{1/2}$, this rewrites $\|Ce\|\cdot \|C^{-1 …
- 109k
1
vote
Upper bound on the size of vectors contained in an ellipsoid?
Denote $|\theta_{[j]}|=s$. Replace $j$ largest (in absolute value) coordinates of $\theta$ to $s$, other coordinates to 0. The vector remains in ellipsoid $\mathcal{E}_D$. If the coordinates equal to …
- 109k
5
votes
Accepted
Inverse of a matrix and the inverse of its diagonals
Let me change the notations and reformulate the problem. You have a positive definite $n\times n$ ($n$ is your $K$) matrix $R$ with diagonal $D$ (your $D$ is $n$ times less than mine), and you have to …
- 109k
19
votes
When the sum of positive definite matrices converges, does the sum of the norm of the associ...
Yes. The norm of a positive definite matrix does not exceed its trace, and the sum of traces is finite, since the sum of diagonal elements is finite for each of $n$ places.
- 109k
3
votes
Bound on the ratio of top 2 eigenvalues
I claim that $\lambda_2'/\lambda_1'\leqslant 1-\frac{2\tau}{1-(n-2)\tau}$ which is bit stronger than you ask for. This is sharp as the example $D_{11}=D_{22}=1\gg \max(D_{33},\dots,D_{nn})$ shows.
Of …
- 109k
4
votes
Non-asympototic version of Gelfand's formula
Yes, this is true for any $c>1$ and large enough $n$.
Let $p(z)=\prod_{i=1}^n (z-\lambda_i)$ be characteristic polynomial of $A$. Denote by $r_k(z)=c_0(k)+c_1(k)z+\dots +c_{n-1}(k)z^{n-1}$ the remai …
- 109k
6
votes
Vandermonde matrix is totally positive
V. Prasolov's book Problems and Theorems in Linear Algebra contains it as Theorem 1.2.12.2.
Russian text in pdf Alas, this very useful book seems to be not translated into English. The previous, trans …
- 109k
3
votes
Accepted
Condition for non-vanishing trace
Your trace equals ${\rm tr} ((BPA^\top+APB^\top)X)$. This equals 0 for all symmetric matrices $X$ if and only if $C=BPA^\top+APB^\top=0$ (else take $X=C$, note that $C$ is symmetric). Of course, this …
- 109k
5
votes
How to prove that each element of $A(A^TA)^{-1}A^Ty$ is greater than 0, if $A(i,j)>0$ and $y...
This does not hold in general.
Note that $\beta=A(A^TA)^{-1}A^Ty\in \mathbb{R}^m$ belongs to the image ${\rm Im} A$ (that is, to the column space of $A$). Also, $A^T\beta=A^TA(A^TA)^{-1}A^Ty=A^Ty$, so …
- 109k