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Results tagged with matrix-analysis
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user 8799
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.
2
votes
Real square roots of symmetric matrices
Eventually, let me present the proof which I mentionned in my comment.
Herebelow, I use repeatedly orthogonally similarities. Mind that if $S'$ is orthogonally similar to $S$, then $S'$ satisfies the …
- 52.3k
4
votes
Accepted
Question on density of certain set of matrices
It suffices to check whether $B^{-1}S=:S'$ has measure zero in $B^{-1}Q=:Q'$. We have
$$Q'={\bf Sym}_n(\mathbb R),\qquad S'=\{{\bf Sym}_n(\mathbb R)|B(\Sigma+\Sigma^3)\in{\bf Sym}_n(\mathbb R)\}.$$
Th …
- 52.3k
9
votes
Accepted
One question on block-circulant matrices
The formula for the specific case is
$$\det K=\det(A+B+C+D)\det(A-B+C-D)\det(A+iB-C-iD)\det(A-iB-C+iD).$$
More generally, for a block-circulant matrix with $n$ square blocks $A_0,\ldots,A_{n-1}$, the …
- 52.3k
0
votes
Cofactor an geometrical mean in $\mathit{SPD}_3$: a Gårding-like inequality
Because of the lack of answers, I continued my investigations, and eventually got it !
Notation : because some expressions are too long for the command widehat, I'll sometimes denote ${\rm Cof}A$ for …
- 52.3k
2
votes
Show that the eigenvalues of a non-symmetric matrix built from positive matrices have positi...
Since there is not much progress on this question, let me give a partial result and a direct consequence.
Denote $S=C-B^TA^{-1}B$ the Schur complement of $A$ in $N$, which is positive definite. Then $ …
- 52.3k
2
votes
Bounding eigenvalue/eigenspace perturbations for hermitian matrices
The eigenvalues are $1$-Lipschitz over the set of Hermitian matrices:
$$|\lambda_k(B)-\lambda_k(A)\|\le\|B-A\|.$$
This Lipschitz property is a very different phenomenon than the analyticity of $t\maps …
- 52.3k
3
votes
Accepted
Operator norm of difference of matrix decompositions
The answer is negative, and this happens as soon as $n=2$. The question is whether the composition $X\mapsto L:=L_{X^2}$ is globally Lipschitz over ${\bf SPD}_n$. Let $x_j\in{\mathbb R}^n$ denote the …
- 52.3k
17
votes
Accepted
Counting eigenvalues without diagonalizing a matrix
Here is an efficient method.
First of all, I must quote that diagonalizing $M$ is not a method, because there is no explicit way to carry this out. It amounts to calculating the roots of a polynomial …
- 52.3k
5
votes
A question of invertibility of matrices
What about
$$A=\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},\qquad B=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\quad ?$$
- 52.3k
2
votes
Accepted
Dubious matrix monotonicity
Such a matrix has the form $\theta I_n+ew^T+ve^T$ where $T$ denotes transposition and $e$ is the vector $(1,\ldots,1)^T$. The parameter $\theta$ is $\le0$.
Here is the proof when $n\ge4$. By continui …
- 52.3k
1
vote
Representation theorem for matrices (reference request)
Here is a down-to-earth proof. Remark that for complex matrices, proving $A=B$ is equivalent to proving $\langle x,Ax\rangle=\langle x,Bx\rangle$ ; the superiority of the complex numbers over the real …
- 52.3k
17
votes
Accepted
The abc-conjecture as an inequality for inner-products?
The matrix $L_n$ is positive definite.
Proof. The matrix $G_n$ with entries ${\rm gcd}(a,b)$ is positive definite because of $G=D^T\Phi D$ where $\Phi={\rm diag}(\phi(1),\ldots,\phi(n))$ ($\phi$ the …
- 52.3k
2
votes
Matrix-convexity of inverse of the cofactor matrix
Well, this is not an answer. But I cannot resist to mention the following equivalent property. Let $A\mapsto \hat A$ denote the cofactor map, and $B\mapsto \check B$ its inverse. For positive definite …
- 52.3k
2
votes
$2$-norm distance between square roots of matrices
I don't have the answer to your question, but I can give you the following:
$$\|\sqrt A-\sqrt B\|_\infty\le\sqrt{\|A-B\|_\infty\,}\,,$$
where the $\infty$-Schatten norm is nothing but the operator nor …
- 52.3k
4
votes
Accepted
Relation between Frobenius norm, infinity norm and sum of maxima
The answer is Yes. This is not really a problem about matrices. The best way to analyse it is to rewrite it in terms of the row vectors $u_i\in{\mathbb C}^n$. Let me denote $\|\cdot\|_p$ the $\ell^p$- …
- 52.3k