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Results tagged with matrix-analysis
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user 8430
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.
6
votes
Accepted
Optimization version of the Sylvester equation
First recall two basic ideas.
Lemma. Let $A$, $B$, $C$ be arbitrary; then, $\text{vec}(ABC) = (C^T \otimes A)\text{vec}(B)$, where $\otimes$ denotes the Kronecker product and $\text{vec}(\cdot)$ d …
- 28.6k
23
votes
Accepted
Does the matrix exponential preserve the positive-semi-definite ordering?
To supplement Robert's counterexample, let me mention below some interesting facts about the matrix exponential, along with what may be regarded as the "correct" way of obtaining matrix exponential li …
- 28.6k
5
votes
Accepted
variation of the Lieb concavity theorem
The conjectured inequality is false.
Using cyclicity of the trace, let's first write it in a slightly nicer form
\begin{equation*}
f(A,B) := \operatorname{tr}(X^*B^{(1-s)/2} A^sB^{(1-s)/2}X).
\end{ …
- 28.6k
2
votes
The eigenvectors and eigenvalues of matrix geometric mean
There are several known relations between eigenvalues of $G$ and $A, M$.
However, as the following equality for $2\times 2$ matrices shows, these relations are as rich as the standard inequalities b …
- 28.6k
1
vote
The derivative of the Cholesky factor
The following may be of help. If $A=T'T$ (where $T$ is upper triangular), then you can show that (see. Thm. 2.1.9 in Aspects of Multivariate Statistical Theory by R. J. Muirhead):
\begin{equation*}
( …
- 28.6k
4
votes
Accepted
norm of (sub)stochastic matrix
A useful and easy to compute bound is given by the reasonably well-known relation (see e.g., this Wikipedia section)
\begin{equation*}
\|A\| \le \sqrt{\|A\|_\infty \|A\|_1}
\end{equation*}
between the …
- 28.6k
1
vote
Perron Frobenius with one negative pair of entries
The following paper (and the large number of references cited therein) provides some general sufficient conditions to ensure the "Perron-Frobenius property," thereby offering a set of useful answers t …
- 28.6k
11
votes
Do singular values dominate eigenvalues?
Let $\lambda(A)$ denote the vector of eigenvalues and $s(A)$ the vector of singular values (arranged in decreasing order). The claim of the question is whether $|\lambda(A)|^{\downarrow} \prec_w s(A)$ …
- 28.6k
9
votes
Accepted
Matrix-convexity of inverse of the cofactor matrix
Not just $3\times 3$, but in general, the map $A \mapsto \det(A^{-1})A$ is operator convex on positive definite matrices.
Proof sketch.
$\newcommand{\pfrac}[2]{\left(\tfrac{#1}{#2}\right)}$
If suffic …
- 28.6k
10
votes
Accepted
Concavity of the trace of a matrix power
Unfortunately, the conjectured function is not concave. Here is a simple simpler counterexample.
\begin{equation*}
B = \begin{bmatrix} 1 & 2 \\ 3 & 4\end{bmatrix},\quad
A = \begin{bmatrix} 2 & 0 \\ …
- 28.6k
7
votes
Accepted
On a determinant inequality of positive definite matrices
A quick counterexample to your conjecture is
\begin{equation*}
A = \begin{pmatrix}
13 & 3 & -13 & -5\\
3 & 4 & -3 & 4\\
-13 & -3 & 13 & 5\\
-5 & 4 & 5 & 10\\ …
- 28.6k
2
votes
Accepted
derivative of sum of singular values
This function is not differentiable (consider $A=0$). If you are interested in learning about its subdifferential (and more on subdifferential of spectral functions), please refer to the excellent pap …
- 28.6k
1
vote
How to compute difference between 2 similarity matrices?
Too long for a comment, but here are a couple of ideas:
Compute distance between the correlation matrices themselves---if your correlation matrices happen to be invertible, then you can use the Riem …
- 28.6k
1
vote
Accepted
Norm/trace of product inequality involving skew symmetric matrices
Something seems to be missing here, because the inequality is trivially seen to be false. Consider the following randomly picked matrices for instance:
\begin{equation*}
B = \begin{bmatrix}0 & -4 & 4 …
- 28.6k
5
votes
Accepted
On proof of the conditionally negative definiteness of a kernel
Here a direct approach. Recall the power-series
\begin{equation*}
\arccos(z) = \frac\pi2 - \sum_{k\ge0}\binom{2k}{k}\frac{z^{2k+1}}{4^k(2k+1)}.
\end{equation*}
From this series it is clear that $\ar …
- 28.6k