All Questions
7 questions
15
votes
3
answers
6k
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Questions on Toeplitz matrices: invertibility, determinant, positive-definiteness
These questions are probably very basic but I'll dare to ask them anyway since I didn't have much luck in Math Stack Exchange.
Let $A$ be an $n \times n$ Hermitian Toeplitz matrix:
$$A = \begin{...
- 3,626
5
votes
2
answers
312
views
minimum-maximum entries matrix
Let $M(n)$ be an $n\times n$ matrix in the variables $x_1,\dots,x_n$ with entries
$$M_{i,j}(n)=\frac{x_{\max(i,j)}}{x_{\min(i,j)}}, \qquad 1\leq i,j\leq n.$$
I'm interested in the following:
...
- 43.2k
4
votes
2
answers
203
views
Results of invertibility of a matrix involving the Szego kernel
In the context of reproducing kernel Hilbert spaces, the Szego kernel is the function $k(z_i,z_j)=\frac{1}{1-z_j\overline{z_j}}$.
Given $2n$ points $\{z_1,\ldots,z_n\},\{w_1,\ldots,w_n\}\in\mathbb{D}\...
- 167
2
votes
2
answers
4k
views
Find inverse and determinant of a symmetric matrix - for a maximum-likelihood estimation
Evaluate the determinant $\det \Omega $ and find the inverse matrix $\Omega^{-1}$ of:
$$\Omega = \begin{bmatrix} \beta_1^2(1+\theta_1^2) & \beta_1 \beta_2 & ... & \beta_1 \beta_{k-1} &...
- 123
2
votes
1
answer
137
views
Existence of matrices with some invertibility properties
Prove that there exists five matrices $B_i \in \mathbb{F}_2^{5\times 10}$, $i\in \{1,2,3,4,5\}$, such that any two $B_i$'s form an invertible matrix in $\mathbb{F}_2^{10\times 10}$.
I am interested ...
user83947
1
vote
1
answer
252
views
Condition on the probabilities for the $J\times J$ matrix $[ \Pr(X=j \mid Y=k) ]$ to be invertible
$\DeclareMathOperator\Pr{P}\newcommand\cPr[2]{\Pr(#1 \mid #2)}$I have a $J \times J$ matrix:
$$
M:= \begin{bmatrix}
\cPr{X=1}{Y=1} & \cPr{X=2}{Y = 1} & \cdots & \cPr{X=J}{Y = 1} \\
\cPr{X=...
- 151
1
vote
1
answer
18k
views
Derivative of log determinant and inverse
I have a matrix $\Sigma$ with element $(i,j)$
$$\Sigma_{i,j}= \exp(-h_{i,j}\rho).$$
The matrix is positive definite and symmetric (it is a covariance matrix).
Now I need to evaluate
$$\frac{\...
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