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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5
votes
1answer
120 views

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 ...
2
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1answer
26 views

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?
4
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2answers
60 views

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 ...
1
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1answer
44 views

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 ...
0
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2answers
86 views

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 ...
5
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2answers
82 views

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 ...
1
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1answer
74 views

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 -> ...
0
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0answers
60 views

Is there an option to handle Neumann-series when it diverges? (using infinite-sized Carleman matrices)

Motivation I'm considering functions, represented by Carleman matrices of infinite size, for instance $f(x)=t^x - 1$. Let us denote the iterates of $f(x)$ by indexes on $x$ like $x_0=x$, $x_1=f(x)$, $...
1
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1answer
68 views

gram matrix for elements of an infinite dimensional hilbert space

I know that a gram matrix is defined as $G = V^*V$ where $G = (\langle x_i , x_j \rangle)_{i=1, j=1}^n$ where $x_i, x_j \in H$ and $H$ is some inner product space, say a Hilbert space. I'm confused ...
5
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1answer
102 views

Find the Inverse of an Infinite Square Matrix

For my mathematics assignment I am using polynomial interpolation to solve certain problems and I end up with the following scenario: $\begin{bmatrix}... & 0 &0 & 0 & 1\\... &1^3 &...
2
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2answers
92 views

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 ...
2
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1answer
93 views

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 ...
0
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1answer
79 views

Inverse of an infinite matrix with factorial entries

Has anyone come across the following matrix? $$ A= \begin{bmatrix} \frac{1}{1!}&\frac{1}{2!}&\frac{1}{3!}&\cdots\\ -\frac{1}{2!}&-\frac{1}{3!}&-\frac{1}{4!}&\cdots\\ \frac{1}{3!...
3
votes
1answer
162 views

Is the Birkhoff–von Neumann theorem true for infinite matrices?

The Birkhoff–von Neumann theorem states that every $n \times n$ doubly stochastic matrix is a convex combination of permutation matrices. Is this true for $\mathbb{N} \times \mathbb{N}$ matrices as ...
3
votes
1answer
104 views

Derive prime-identifying functions from inverse Vandermonde and Bernoulli numbers

From the identity $\ln\zeta(s) = \frac{q(1)}{1^{s}}+\frac{q(2)}{2^{s}}+\frac{q(3)}{3^{s}}+\frac{q(4)}{4^{s}}+\ldots$, where $q(n)= \begin{cases} \frac{1}{k} & \text {$n={p}^{k}, k \in \mathbb{N} ...
0
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0answers
46 views

Distance from a vector to a linear span vectors in a separable Hilbert space.

Consider a separable Hilbert space $\mathcal{H}$. Let $F$ be a vector subspace of $\mathcal{H}$ and let $v\in\mathcal{H}$. If $F$ is of finite dimension $n$, then the distance from $v$ to $F$ can be ...
4
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2answers
127 views

Infinite matrix which cannot be represented by bounded linear operator

Let ${\cal V}$ be n Hilbert space over $\mathbb{R}$ or $\mathbb{C}$, with an orthonormal basis $(e_n)_{n=1}^{\infty}$. For every bounded linear operator $A: {\cal V} \to {\cal V}$, we can associate $A$...
1
vote
1answer
124 views

Spectrum of an infinite matrix

I'm currently reading the article "Full Banach Mean Values On Countable Groups" by Harry Kesten (see http://www.mscand.dk/article/view/10568/8589) for my bachelor's thesis. For a given matrix $M$, ...
9
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0answers
148 views

Matrix exponential, containing a thermal state

Define an infinite matrix $$ M = \begin{bmatrix} 0 & -1 & 0 & 0 & \cdots \\ 1 & 0 & -2 & 0 & \cdots \\ 0 & 2 & 0 & -3 & \cdots \\ 0 & 0 & 3 &...
2
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1answer
138 views

Inverse of an Infinite Matrix (with factorials)

How to calculate this monstrous expression? $$ \begin{pmatrix} \frac{1}{1!} & \frac{1}{2!} & \frac{1}{3!} & \frac{1}{4!} & \frac{1}{5!}& \cdots\\ 0 & \frac{1}{1!} & \frac{...
3
votes
2answers
124 views

How can I find the inverse of this infinite triangular matrix?

I want to find the inverse of the following matrix \begin{bmatrix} 1&0&0&0&0&\cdots&0&0&\cdots\\ 0&1&0&0&0&\cdots&0&0&\cdots\\ \binom{2}{...
2
votes
2answers
384 views

Is infinite upper triangular matrix with nonzero entries on the diagonal invertible?

Let's say we have an infinite square matrix where rows and columns are indexed by $\mathbb{N}$. We also know that every entry on the diagonal is nonzero and the matrix is upper triangular. Is it ...
3
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0answers
234 views

Finding all eigenvalues of a bounded strictly upper triangle operator in a Hilbert Space

I've been working on this question for awhile, and am not sure how to finish the argument. It feels like I'm on the right track though. I've demonstrated my approach below: Question: Let $E$ be a ...
1
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1answer
34 views

Decomposing $M_n(R)$ and $T^+_n(R)$ into a direct sum of their rows

I have been asked to show both $M_n(R)$ (the $n\times n$ matrices over a ring $R$) and $T^+_n(R)$ (the $n\times n$ upper triangular matrices over $R$) both decompose into a direct sum of their rows (...
5
votes
1answer
85 views

If $\sum\limits_i\sum\limits_j\alpha_{ij}x_iy_j$ converges for every square integrable $(x_n)$ and $(y_n)$, then the order of the sums commutes

Let $(\alpha_{ij})$ be a square infinite matrix such that for all $x=(\xi_{n}),y=(\eta_{n}) \in \ell ^{2}$ we have that ${\displaystyle \sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\alpha_{ij}\xi_{i}\eta_{j}}...
1
vote
1answer
191 views

A question about the cyclic property of traces for infinite dimensional matrices

Does the cyclic property of trace hold for infinite dimensional matrices (which are often encountered in quantum mechanics)? In other words, does the identity $$\rm Tr(ABC)=Tr(CAB)=Tr(BCA)$$ hold when ...
0
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1answer
284 views

Limiting distribution of an infinite Markov Chain

Let the following infinite matrix P represent an Infinite States Markov Chain. \begin{pmatrix} 1 & 0 & 0 & 0 & \cdots\\ 1 & 0 & 0 & 0 &\cdots\\ 0 & 1 & 0 &...
1
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0answers
73 views

Infinite matrix of non negative numbers (sequence)

Suppose that we have matrix $\begin{Bmatrix}a_{ij}\end{Bmatrix}_{n = 1}^{\infty}$ of non negative real numbers. Let $\pi : \mathbb{N}\rightarrow \mathbb{N} \times \mathbb{N}$ be bijection so $\begin{...
0
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1answer
58 views

Linear Transformations in Infinite Spaces

While studying for Linear Algebra, I stumpled upon two questions, both formatted the same, yet one being more difficult for me than the other. Question A Given a finite dimensional linear subspace $...
1
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0answers
376 views

Solving an infinite system of linear first order differential equations (homogeneous)

I have a countable infinite set of functions $\{\Gamma_n(\alpha),n\in\mathbb{N}\}$ over the interval $0<\alpha<1$. They should satisfy a linear system of differential equation (first order,...
13
votes
1answer
456 views

Do eigenvalues of a linear transformation over an infinite dimensional vector space appear in conjugate pairs?

While attempting to answer a question here (namely, the finite dimensional case of the title question: Prove that if $\lambda$ is an eigenvalue of $T$, a linear transformation whose matrix ...
3
votes
0answers
53 views

Hermite normal form of infinite matrix

I have an infinite matrix $A$ that is in row echelon form, except zeros are to the right of each pivot, as shown below. The $*$ symbol denotes any integer, though most of them are zero. $A= \left( \...
3
votes
1answer
582 views

Infinite matrix product

Let $$X=\left(\begin{array}{c} x_1 \\ x_2\\ \vdots \end{array}\right)$$ be an infinite real vector and $$A=(a_{ij}), \ 0<i,j<\infty$$ be an infinite real matrix. (1) For which $A$ can one ...
8
votes
2answers
2k views

How can I get eigenvalues of infinite dimensional linear operator?

What I want to prove is that for infinite dimensional vector space, $0$ is the only eigenvalue doesn't imply $T$ is nilpotent. But I am not sure how to find eigenvalues of infinite dimensional linear ...
2
votes
0answers
208 views

Solving recursions by calculating determinant of an infinite matrix

In this reference (pg. 4) and few others a specific parameter in a recursion formula is solved by setting the determinant of an infinite matrix to $0$. In this precise case we have $c_{n-1} - D_n c_n ...
2
votes
3answers
10k views

How do i find the determinant of this nxn matrix?

I have the following matrix with dimensions nxn: $$ \begin{bmatrix} -2 & x & x & \cdots & x \\ x & -2 & x & \cdots & x \\ x & x &...
1
vote
4answers
151 views

Consider the following system and find the values of b for which the system has a solution

So I have this system: $$\left\{\begin{array}{c} x_1 &− x_2 &+ 2x_3 &= 2 \\ x_1 &+ 2x_2 &− x_3 &= 2 \\ x_1 &+ x_2 & &= 2 \\ x_1 & & +x_3 &...
1
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0answers
82 views

$det(I+A(\epsilon))$ where $A$ is an infinite matrix and not trace class!

Assume that $A$ is an infinite matrix and it's a function of the parameter $\epsilon$. I would like to find $\epsilon$ so that the $det(I+A(\epsilon))=0$. I know if $A$ was a trace class I could use ...
2
votes
2answers
312 views

Confusion between spectral radius of matrix and spectral radius of the operator

The adjacency matrix $A(G)$ of an infinite undirected graph $G$ is considered as a bounded self-adjoint linear operator $A$ on the Hilbert Space $l^2(G)$ (last section of https://en.wikipedia.org/wiki/...
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0answers
370 views

What properties do the rings of infinite, upper-triangular matrices have?

I'm very familiar with the ring of $n\times n$ matrices over a field, the ring of $n\times n$ upper triangular matrices over a field, and the ring of infinite column-finite matrices over a field. But ...
3
votes
1answer
118 views

Stability for a infinite dimensional dynamical system

Suppose I have a infinite dimensional dynamical system as $\dot{x_n}=Ax$ where $A$ is an infinite-dimensional matrix and $\{x_n\}$ is a sequence of states. I was wondering if you can help me find ...
3
votes
2answers
213 views

Infinite Matrices

Does anyone know how to prove that the set of all $K\times K$ column finite matrices over a ring $R$ $[\mathrm{CFM}(R)]$ forms a ring? I am also confused about the definition of multiplication in $\...
1
vote
1answer
129 views

Inverse of Infinite Block Matrix

Let $B, C$ be two $2\times 2$ matrices with complex entries. Let an infinite block matrix $H$ be given by its component: $H_{n,n'} = B \delta_{n,n'+1} + B^\dagger \delta_{n,n'-1} + C \delta_{n,n'} $. ...
1
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0answers
81 views

Is there a systematic relation between the generating functions for the rows vs that for columns of infinite sized Carleman-matrices?

(Roughly related, but generalizing, of this earlier question) Background: The first part of the following(the column-wise-focus) is also described in Eri Jabotinski's 1953-treatize Representation of ...
5
votes
1answer
153 views

Why is it so difficult to obtain the spectral properties related to infinite matrices, especially when they are not symmetric?

I believe most of the spectral theory is revolving around the bounded self-adjoint linear operators being analogous to real symmetric infinite matrices. Whereas, there are cases when the matrices are ...
3
votes
2answers
895 views

Inverting the infinite matrix $+\mathbf{I}$ with entries $\mathbf{P}_{ij}={i-1\choose j-1}$ [closed]

Let $ \mathbf{P}$ denote the "infinite matrix" $$ \left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & \dots \\ 1 & 1 & 0 & 0 & \dots \\ 1 & 2 & 1 & 0 & \dots \...
2
votes
1answer
171 views

Orthogonal Operator Infinite Dimensional Inner Product Space

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 $\...
2
votes
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 &...
1
vote
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\...
0
votes
2answers
66 views

how to compare two infinite dimensional operators

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 ...