All Questions
12 questions
1
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0
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29
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Can all real positive semidefinite Hankel matrices be decomposed into sum of rank 1 real positive semidefinite Hankel matrices?
Denote the set of real positive semidefinite $d\times d$ Hankel matrices as $\mathcal{S}$. Can we always decompose one $S\in \mathcal{S}$ into sum of rank $1$ $S_i\in \mathcal{S}$, i.e., $S=\sum_i S_i$...
- 11
8
votes
1
answer
325
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On a matrix inequality
$\newcommand{\R}{\mathbb R}\newcommand{\tr}{\operatorname{tr}}$It follows from Proposition 7 and this recent answer that, for any positive-definite $n\times n$ symmetric real matrices $A$ and $B$,
$$\...
- 128k
4
votes
1
answer
408
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Proof of Levinson-Durbin algorithm
Is there any article or reference book with a full proof of the Levinson-Durbin algorithm used for solving linear system with a Toeplitz matrix ?
- 41
5
votes
2
answers
634
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Reference request: continuity of Cholesky factor
It most books dealing with Cholesky decomposition, or it is variants, one finds a statement of the form if $A$ is symmetric $k\times k$ positive semi-definite (non-negative definite) then the $k\times ...
- 5,405
3
votes
1
answer
421
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Inequality for $AB + BA$ when $A,B\geq0$, reference request
Let $A,B\geq0$ be positive semidefinite matrices of arbitrary size $n\times n$. Denote by $\alpha$ and $\beta$ their largest eigenvalues.
It is well-known that the eigenvalues of the expression $AB +...
- 219
8
votes
3
answers
663
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Representation theorem for matrices (reference request)
Motivation. If $A \in \mathbb{C}^{n \times n}$ is self-adjoint (or, more generally, normal), then we all know that
$$
A = \sum_{k=1}^n \lambda_k \, h_k \otimes h_k,
$$
where $\lambda_1,\dots,\lambda_n$...
- 12.6k
14
votes
4
answers
3k
views
Vandermonde matrix is totally positive
A totally positive matrix $M\in \mathcal{M}_{n\times m}(\mathbb R)$ is such that all of its minors of all sizes are positive. It is true that any Vandermonde matrix (with well-ordered positive entries)...
- 5,428
4
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0
answers
245
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On the sum of the first row of the inverse of a certain symmetric Toeplitz matrix
(i) Consider a Toeplitz matrix $A_n = (a_{i, j})_{1 \le i, j \le n}$ of size $n$ defined as follows:
$$ a_{i, j} := |i-j|^{-1/2}, \text{ if } i \ne j; \ \ a_{i, j} := 2, \text{ if }i = j. $$
Let $...
- 195
2
votes
0
answers
275
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Maximum spectral norm of matrices with given anti-Hermitian part and Hermitian part's spectrum
Let $M\in M_n(\mathbb C)$ be a $n\times n$ matrix over the complex field. It can be written uniquely as $M=H+A$, where $H=H^*$ denotes its Hermitian part and $A=-A^*$ its anti-Hermitian part.
Its ...
- 21
26
votes
1
answer
1k
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Real square roots of symmetric matrices
In joint work with Andreas Fischle (TU Dresden, Germany) and Patrizio Neff (U Essen, Germany) we needed to use the following statement: If $S$ is a real $n\times n$ matrix with $S^2$ symmetric, then ...
- 5,186
3
votes
2
answers
940
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Positive definiteness of infinite tridiagonal matrices
I am interested in the following problem: I have an infinite symmetric tridiagonal matrix
$$
A=
\begin{bmatrix}
a_1 & b_1 & & & \\
b_1 & a_2 & b_2 & & \...
- 692
1
vote
0
answers
90
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Algorithms to compute the rank of a parametrized matrix [closed]
Motivated by my question on Mathematics StackExchange and by a question by Anirbit on the same site, I ask for some references on the problem of rank computation for a parametrized matrix. References ...
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