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Results tagged with matrices
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user 56553
Questions where the notion matrix has an important or crucial role (for the latter, note the tag matrix-theory for potential use). Matrices appear in various parts of mathematics, and this tag is typically combined with other tags to make the general subject clear, such as an appropriate top-level tag ra.rings-and-algebras, co.combinatorics, etc. and other tags that might be applicable. There are also several more specialized tags concerning matrices.
7
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1
answer
375
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Hankel matrix commuting with a Jacobi matrix
For instance, interesting Hankel matrices correspond to the choice
$\alpha_{n}=1/n^{2}$ (or more general powers of $n$) or $\alpha_{n}=1/n!$.
Any information related to the post would be useful. …
- 2,188
2
votes
1
answer
84
views
Information on special matrices similar to Jacobi matrices
Jacobi matrices are well known and deeply investigated mathematical objects from various point of view. … So far, I did not find any literature devoted to the study of mathematical properties of these matrices. I am interested in spectral properties of these matrices, in particular. …
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2
votes
Accepted
The norm of a Finite Hilbert matrix
I think the answer is no, indeed. Even in the particular case $p=2$ the formula is not known (to my best knowledge). Concerning $\|H\|_{2}$, is known $\|H\|_{2}\leq\pi$. More precisely, we have the fo …
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1
vote
2
answers
218
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Norm of a matrix operator with a special structure
Let $\{\alpha_{n}\}_{n\in\mathbb{N}}$ be positive sequence such that
$$\sum_{n=1}^{\infty}\alpha_n<\infty.$$
Question: Is there any chance to evaluate the operator norm of the matrix operator
$$C=\b …
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4
votes
0
answers
147
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A Toeplitz variant of the Hilbert matrix
It is well-known that the Hilbert matrix $H$, i.e., the symmetric Hankel matrix with entries
$$H_{m,n}=\frac{1}{m+n-1}, \quad m,n\in\mathbb{N},$$
determines a bounded operator on $\ell^{2}(\mathbb{N …
- 2,188
7
votes
1
answer
1k
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Origins of the Jacobi matrix
I have several questions concerning history of Jacobi matrices.
Does anybody know why the Jacobi matrix (=symmetric tridiagonal matrix) is named by Carl Gustav Jacob Jacobi? … What is the origin and the history of methods of the investigation of spectral properties of Jacobi matrices?
Any suitable reference concerning the above questions would be helpful. Thanks. …
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1
vote
Simple Spectrum of Jacobi matrices
In the case of finite or semi-infinite Jacobi matrix, the first entry of an eigenvector uniquely determine other entries since they are related by the tree-term recurrence. This is the reason why the …
- 2,188
0
votes
Eigenvalues of symmetric tridiagonal matrices
Withnout loss of generality, one can put $a=0$. For sure, there is no closed-form (or explicit) formula for the eigenvalues in general. However, at least the characteristic polynomial of $A_n$ can be …
- 2,188
3
votes
2
answers
308
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A relation between norm and spectral radius for some matrix operators on Banach spaces $\ell...
Let $A=(a_{i,j})_{i,j=1}^{\infty}$ be a semi-infinite matrix with real entries. Suppose further that $A$ and $A^{T}$ (matrix transpose) represent bounded operators on $\ell^{p}$ for $p\geq1$. Denote f …
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6
votes
Is $1/\max(i,j)$ a bounded matrix on Hilbert spaces?
The operator you introduced, say $A$, is bounded indeed. There is a simple proof for this:
First, note $A$ can be written as $A=CC^{*}$ where
$$C_{i,j}=\begin{cases}
\frac{1}{i}, & i\geq j,\\
0, & i …
- 2,188