# Questions tagged [infinite-matrices]

For questions involving matrices of infinite size, often identified with bounded linear operators on infinite-dimensional separable Hilbert spaces.

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### How to invert $\infty \times \infty$ matrix?

I have an equation $AX=B$, where $A$ is $\infty \times \infty$ matrix, $X$ is $\infty \times 1$ vector and $B$ is $\infty \times 1$ vector. $A$ and $B$ are known and I need to determine $X$. For ...
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### Can a matrix have an uncountably-infinite( aleph-one or aleph-two) dimensions?

I have heard of infinite-dimensional matrices that have a countably-infinite number of dimension. Is it possible that there could be a matrix with aleph-1, aleph-2, or even aleph-aleph-0 dimensions?
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### Prove that when writing up all even numbers in a column, then chaining $2n+1$ from them, we get all natural numbers exactly once.

Here is the idea: Write up all even numbers in the first column, then get numbers in the second column by taking the number to the left ($n$), and calculating $2n+1$. And keep repeating this. Here is ...
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### System of infinitely many linear equations [closed]

I have a system of infinitely many linear equations, where each equation is in $n$ variables, whose solution the trivial solution. How can I show that a finite subset of those equations also only have ...
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### Is real power of unitary matrix unitary?

Suppose $X$ is a unitary matrix. Would $X^k$ also be unitary, where $k \in \mathbb{R}$ (negative as well)? What if $X$ has infinite dimension? I think for $k \in \mathbb{Q}$ the ...
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### Can we approximate any eigenvalue of an infinite matrix via eigenvalues of some sequence of submatrices which approximates the matrix?

Let $T:\ell^2\to\ell^2$ be a compact linear operator. Let $[T]=(a_{i,j})_{i,j=1}^{\infty}$ be the representing infinite matrix of $T$ with respect to the canonical base. Let $T_n$ be the finite rank ...
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### Vandermonde infinite matrix inverse

I am searching for an inverse of a certain infinite matrix, Vandermonde one. I have been searching in bibliography and some well known examples exist in literature: Pascal Matrix Inverse -> ...
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### Can infinite matrices represent nonlinear operators?

I read this reddit post and this SE thread discussing how to represent nonlinear/linear transforms in matrix notations but they were not sufficient. In quantum mechanics, scientists use infinite ...
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### Cutting off an infinite matrix(Making a finite matrix from an infinite matrix)

Let's say that there is an infinite matrix A. How do you make a new finite matrix B, from the matrix A? What I mean is that I want to get a finite matrix that best matches/approximates the original ...
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I know that on a finite dimensional inner product space, a unitary (or orthogonal) operator preserves the inner product. That is, having $\|T(x)\|=\|x\|$ for all $x\in V$ is equivalent to having $\... 0answers 501 views ### Maximum eigenvalue and a corresponding eigenvector of an infinite Hilbert matrix I have the following matrix$$H=\begin{bmatrix} 1 & \frac{1}{2} & \cdots & \mbox{ad}\ +\infty\\ \frac{1}{2} & \frac{1}{3} & \cdots & \mbox{ad}\ +\infty\\ \vdots & \vdots &... 1answer 270 views ### Frechet derivative of shift operator in$l_2$? Let$x \in l_2$and$J(x) = \sum_{k = 1}^{+\infty} x_k x_{k + 1}$. Find$DJ(u)$and$D(DJ(u))$. Attempted solution Since$x \in l_2$, then$\sum_{k = 1}^{+\infty}x_k < \infty$. Another fact:$\|x\...
Perturbation theory is crucial in quantum mechanics. I was wondering how to compare two operators in the case of two infinite-dimensional operator? Assume we have Hamiltonian $H_0$ and $V$. How ...