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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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Is the category of vector spaces with row-finite linear maps an abelian self-dual category?

Fix a field $K$. Given a vector space $V$ with a basis $B$, a vector space $W$ with a basis $C$ and a linear map $f: V \to W$, let $\{f_{c, b}\}_{c, b}$ be the representing matrix of $f$, meaning that ...
Smiley1000's user avatar
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0 answers
27 views

When is a Symmetric Block Toeplitz Matrix Invertible?

Let $$ Q = \begin{bmatrix} Q_0 & Q_1 & Q_2 & \cdots\\ Q_{-1} & Q_{0} & Q_1 & \cdots\\ Q_{-2} & Q_{-1} & Q_0 & \cdots\\ \vdots & \vdots & \vdots & \ddots ...
Benjamin Tennyson's user avatar
1 vote
0 answers
50 views

Eigenvalues of a Hollow Tridiagonal Matrix in Finite and Infinite Dimensions

I am currently studying a system characterized by a particularly structured matrix, as shown below, $$ \dot{ \boldsymbol{x}} = \begin{pmatrix} 0 & -2c & \\ c & 0& 3c & \\ &-2c &...
Ba_nanza's user avatar
  • 138
1 vote
1 answer
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Can I perform finite-dimensional inner products with functions as vector components?

I'm an engineer. I'm struggling with infinite-dimensional linear spaces and would appreciate some help. I will describe my question as an example because I'm unfamiliar with the subject. Let's say $u(...
Marcelo's user avatar
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2 votes
0 answers
82 views

What is the most efficient way to find the coefficient before the $n$-th term in this generating function?

I am calculating the square of the Pascal matrix (i.e. the infinite lower triangular matrix with entries $A_{nk}={n\choose k}$) using the theory of Riordan arrays, and have obtained (or so I believe) ...
Daigaku no Baku's user avatar
1 vote
1 answer
68 views

Inequality on infinite matrices

Take the square of sum of elements in every column for first finite rows (say m). Then take their sum for all the columns (say n). If $m,n \rightarrow \infty$, can we show that it is bounded by some ...
user107723's user avatar
1 vote
1 answer
122 views

A infinite system of linear equations

Suppose that for all $k\in\mathbb{N}$ we have $$\sum_{i=-\infty}^{\infty}a_{ik}x_i=e_k$$ for the complex numbers $a_{ik}$, $e_k$. Let $A$ be the infinite matrix whose rows are the infinite vectors $$(\...
LK_'s user avatar
  • 31
10 votes
3 answers
1k views

Derivative as a matrix: $\mathbf{D}=\dfrac{\mathrm{d}}{\mathrm{d}x}$

I have a strange question, it is possible to consider the derivative as a matrix? (Both are linear transformation technically). I thought about this example, since $1, x,x^2,...,x^n$ can be thought of ...
Math Attack's user avatar
1 vote
1 answer
47 views

When the number of matrices become infinity in a matrix inequality?

I will first show my matrix inequality, and state my confusion after it. Suppose we have $n$ positive semi-definite matrices $M_i$ which further satisfy $\sum_{i=1}^n{M_i}=I$. The dimension of $M_i$ ...
narip's user avatar
  • 67
1 vote
0 answers
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finding stationary solution of a continuous time markov chain.

With a certain rate $R$ balls fall into a box. There is no limit to the number of balls the box can hold, but each ball has a rate $\gamma$ to leave the box and when two balls hit each other they ...
bawo__'s user avatar
  • 21
2 votes
2 answers
296 views

Under what conditions is an infinite matrix a bounded linear transformation in $\ell^2$?

Let $(e_n)_{n\in\mathbb N}$ be the canonical basis of $\ell^2$, and $\mathcal L(\ell^2)$ be the set of bounded linear transformations from $\ell^2$ to itself. If $A\in\mathcal L(\ell ^2)$ we can set $$...
Derivative's user avatar
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0 votes
1 answer
79 views

One-dimensional integral of an exponential integrand which is a function of an infinite-dimensional operator

I'm hoping someone can help me. I'd ideally like to find a closed form for the following integral: $$\text{N} = \int_{0}^{\infty} r e^{-c(rI + A)^2} dr$$ where $r,c \in \mathbb{R}$, $c > 0$, $I$ is ...
Roger Milbertson's user avatar
2 votes
1 answer
82 views

Proof in infinite-dimensional space

Let $a \neq b\neq c \neq a$ be distinct real numbers, and let $f\colon E \to E$ be an endomorphism of a real vector space $E$ such that $$ (f − aI)(f − bI)(f − cI) = 0. $$ Show that $$ E = \ker(f − aI)...
Carinha logo ali's user avatar
2 votes
1 answer
75 views

Do infinite state Markov chains have eigenvalues?

A few days ago, I saw a post showing how the stationary distribution of an infinite-state Markov chain can be computed. Since for a finite-state Markov chain, we can solve its eigenvalue equation to ...
Jack's user avatar
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1 vote
0 answers
50 views

Block diagonalization of infinitely sparse matrix

I am reading the following paper about multi-mode Floquet theory. They have the following matrix $H_F$ given with entries $({H_F})_{nk} = (E+n\omega) \delta_{n,k}+\sum_{i=1}^2 V^i \left( \delta_{n-k,...
Dave Force's user avatar
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0 answers
45 views

Good introduction to infinite dimensional linear systems?

I'm searching for some introductory material to infinite-dimensional linear systems, but on the web I'm running into mainly control theory references and things about applied mathematics and/or ...
marco trevi's user avatar
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1 vote
1 answer
179 views

Matrix representation of operators on $\ell^2$.

Let $A=(a_{nk})_{n,k=1}^{\infty}$ be an infinite matrix that defines a bounded operator on $\ell^2(\mathbb{N})$ by matrix multiplication. Suppose that $B=(b_{nk})_{n,k=1}^{\infty}$ satisfies $|b_{nk}|\...
Jake28's user avatar
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1 vote
0 answers
34 views

Please give examples of countably-infinite matrices that are Hermitian &/or invertible, & that correspond to operators commonly encountered in physics

I'm new to quantum mechanics, and I believe that understanding infinite vector spaces should help my understanding of the math involved. Most of the time I'm only dealing with infinitely ...
Ariel Hernandez's user avatar
1 vote
0 answers
84 views

Infinite matrix such that $|a_{i,j}| \leq r^{|i-j|}$

Let $A=(a_{i,j})_{i,j \in \mathbb{N}}$ be an infinite matrix. Assume there exists $r < 1$ such that $$|a_{i,j}| \leq r^{|i-j|} , \forall i,j \in \mathbb{N}$$ Show that A define a bounded operator ...
Théo Pontasse's user avatar
3 votes
2 answers
164 views

nullity of a random matrix (infinite dimension) in $L^2$ space \begin{pmatrix} 1 & a & & \\ & 1 & a & \\ & & 1 & \dots\\ & & & \dots \end{pmatrix}

This problem comes from condensed matter physics. Matrices are acting on $L^2$ space, since wavefunctions are normalizable. Right, left eigenvectors are a pair of Majorana representation of Fermions, ...
Jian's user avatar
  • 285
1 vote
0 answers
533 views

Traces of infinite-dimensional matrices

Suppose $V$ is an infinite-dimensional vector space (dimension $N$ over the field $k$), and suppose $A$ is an $(N \times N)$-matrix over $k$ which defines a bijective linear transformation $f$ on $V$. ...
Boccherini's user avatar
1 vote
1 answer
123 views

Prove that, if $A, B$ are normed vector spaces then $(A\times B, \|\cdot \|)$ is a normed vector space.

I would like to understand what I must prove. Can you help me with this? Question: Prove that, if $A, B$ are normed vector spaces then $(A\times B, \|\cdot \|)$ is a normed vector space. Note: Note ...
user avatar
2 votes
0 answers
241 views

What is the smallest eigenvalue of this infinite, symmetric, tridiagonal matrix?

$A_{nm}$ ($n,m = 0, 1, 2, \ldots$) is a symmetric, tridiagonal matrix. The diagonal elements are $A_{nn} = a_n = n + 1$, and the off-diagonal elements are $A_{n,n+1} = A_{n+1,n} = b_n = \lambda \sqrt{\...
user1764468's user avatar
0 votes
1 answer
47 views

The limit of finite dimensional linear system of equations

Consider the multiplication operator $$K_n:L^2(0,1)^n \rightarrow L^2(0,1)^n, \\ V(x)=(v_1(x),...,v_n(x)) \mapsto A_nV(x) ,$$ where $n$ is a positive integer and $A_n$ is a constant matrix of size $n$....
Gustave's user avatar
  • 1,523
1 vote
0 answers
171 views

Exponential of infinite dimensional matrix

Asked this on MathOverflow with some more details I have a matrix originating from Master Equation for birth death process on semi infinite lattice. It is tridiagonal. It is not symmetric. I wanted ...
plambda's user avatar
  • 11
0 votes
1 answer
136 views

Are there infinite involutory matrices?

This is really a 3 part question:- Are there infinite involutory matrices of a given order $n\times n$? This article claims that there are infinite involutory matrices, but I think it only claims ...
napstablook's user avatar
4 votes
0 answers
232 views

Spectrum of semi-infinite Toeplitz matrices

I am considering a self-adjoint semi-infinite Toeplitz matrix, by which I mean $$M = \left(\begin{array}{ccccc} a_0 & a_1 & a_2 & a_3 & \cdots \\ a_1^*& a_0 & a_1 & ...
Tomáš Bzdušek's user avatar
1 vote
0 answers
211 views

Finding the eigenvalues of a discrete laplacian on an infinite lattice

If we define the Laplacian as a square matrix with zeroes on the diagonal, and $-1$ on the diagonals exactly above and below the main diagonal, and $0$ everywhere else, how would one go about finding ...
pyroscepter's user avatar
1 vote
0 answers
39 views

Existence of ultrafilter on the positive integers

Let $(a_{n,k}: n,k \ge 1)$ be an infinite matrix of positive reals such that $\sum_{k\ge 1}a_{n,k}=1$ for all $n\in \mathbf{N}$. Question. Is it true that there exist a free ultrafilter $\mathscr{F}$ ...
Paolo Leonetti's user avatar
1 vote
1 answer
58 views

Existence of ultrafilter and real sequence

Let $(a_{n,k}: n,k \ge 1)$ be an infinite matrix of positive reals such that $\sum_{k\ge 1}a_{n,k}=1$ for all $n$. Question. Is it true that there exist a free ultrafilter $\mathscr{F}$ on $\mathbf{N}...
Paolo Leonetti's user avatar
1 vote
0 answers
143 views

Infinite dimensional matrix on countable basis vector space having uncountable eigenvalues?

In my last question on this site (eigenvalues of operator with strange commutator), I found a matrix $A+B$ with the commutator $$[A,B]=\lambda (A+B)$$ Where $A_{nm}=kn(\delta_{nm}+\delta_{n,m+1})$ and ...
fewfew4's user avatar
  • 881
2 votes
1 answer
344 views

Operator norm of an infinite diagonal matrix

Say, $D=\operatorname{diag}(d_1,d_2,d_3,\ldots )$ is an infinite diagonal matrix. I was able to prove that $$\|D\|\leq \sup_{k\geq 1}\{|d_k|\}.$$ Question: Does equality hold? I can see how trivial ...
chhro's user avatar
  • 2,266
2 votes
1 answer
208 views

A guess of linear map on infinite dimensional space.

Let $V$ be a countable infinite dimensional linear space, $$ e_1, e_2, e_3, \ldots $$ are the basis of $V$. $$ f: V \rightarrow V $$ is a left invertible linear map. For any $e_i$, assume $$ f(e_i)= ...
Enhao Lan's user avatar
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1 vote
2 answers
418 views

When is a series of matrices divergent. How to define divergence in this case?

In Quantum Mechanics we deal with series of operators represented as matrices like $$e^A = 1+ A + \frac{A^2}{2} + \dots$$ and similarly for $\sin(A) $, etc., where $A$ is a matrix. Now my question ...
Shashaank's user avatar
  • 903
1 vote
0 answers
194 views

Associativity of infinite matrix product.

Many texts reads "It is well known that for infinite matrices multiplication is non-associative". A treatise on this can be found in On the associativity of infinite matrix multiplication. However, ...
Dr Potato's user avatar
  • 812
7 votes
3 answers
3k views

Invertibility of infinite-dimensional matrix

I have a matrix $M \in \mathbb{R}^{n \times n}$ whose columns are linearly independent. Hence, $M$ is invertible. How to extend this conclusion to the case where $n$ is infinite? Specifically, ...
lulu's user avatar
  • 111
1 vote
0 answers
111 views

Eigenvalues of an infinite block upper triangular matrix

Suppose that I have a compact operator on a separable complex Hilbert space $T:H \to H$. Furthermore suppose that with respect to a suitable orthonormal basis $(e_n)_{n\in\mathbb Z}$, the matrix of $...
Good Boy's user avatar
  • 2,210
0 votes
0 answers
83 views

Approximate the eigenvalues of a diagonalizable infinite matrix by its truncations

Consider the compact operator $ T: \ell^2(\mathbb{N}) \rightarrow\ell^2(\mathbb{N})$, represented by the infinite matrix $(a_{i j})_{i,j=0}^\infty$ in the canonical Hilbert basis $(\delta_i)_{i=0}^\...
KRPO's user avatar
  • 33
0 votes
0 answers
35 views

Factorize singly infinite singular matrix to shifting matrix

Claim: For any singly infinite non-invertible matrix $A$, let $A$ to be injective and $A=BC$, where $B$ is invertible, and $C$ is a product of shifting matrices. Is this claim true? Any reference ...
Chengpei's user avatar
3 votes
0 answers
91 views

Are involutions in infinite vector spaces always diagonalisable?

As the title says, I'm wondering if involutions in infinite dimensional vector spaces always have the entire space as its eigenspace? This is of course true for finite vector spaces, but I've never ...
whyamidoingthis's user avatar
1 vote
1 answer
112 views

What does "matrix $A$ induces a linear operator on $\ell^2$" mean?

In the description of Toeplitz matrices, it is said that: A bi-infinite Toeplitz matrix $A$ induces a linear operator on $l^2$. I do not know what does it mean to a matrix induce an operator. And ...
strahd's user avatar
  • 111
1 vote
1 answer
115 views

Which of the statements is/are TRUE?

Let $K = \big [k_{i,j} \big ]_{i,j = 1}^{\infty}$ be an infinite matrix over $\Bbb C$ (the set of all complex numbers) such that $(\text {i})$ for each $i \in \Bbb N$ (the set of all natural ...
math maniac.'s user avatar
  • 2,013
2 votes
0 answers
136 views

Can I assign the term "is eigenvector" and "is eigenmatrix" of matrix **P** in my specific (infinite-size) case?

Background: I'm rereading a couple of my exploratory math-manuscripts and want to fix some possible wrong or misleading expressions. I've used the following notions/expressions a couple of years ...
Gottfried Helms's user avatar
0 votes
1 answer
46 views

If $I+B$ is invertible in $l^2(\mathbb{Z})$ where $B$ is a compact opreator, is $I+B^N$ invertible where $B^N$ is a truncated version of $B$?

Let $I$ be the identity operator in $l^2(\mathbb{Z})$ and let $B$ be a compact operator in $l^2(\mathbb{Z})$ and suppose $I+B$ is invertible. $B$ can be written as a bi-infinite matrix with elements $...
sonicboom's user avatar
  • 10k
6 votes
1 answer
764 views

Does Schmidt decomposition exist on infinite dimensional Hilbert spaces?

A pure quantum state in a bipartite system, which is an operator $$\rho = \langle\psi \,,\, \cdot \,\rangle \, \psi \in \mathcal{L}(H_1 \otimes H_2)$$ for some $\psi \in H_1 \otimes H_2$, is ...
Sahdo's user avatar
  • 85
2 votes
4 answers
175 views

Matrixes of higher order like $M_{\aleph\times \aleph}$ [closed]

I was wondering if thinking about matrixes of the form $M_{\aleph\times \aleph}$ and assigning properties to them like matrix multiplication makes sense and useful in some way. note: $\aleph$ is the ...
David.S's user avatar
  • 133
0 votes
0 answers
310 views

reduced row echelon form for an infinite matrix

It is well-known that the reduced row echelon form for a finite matrix is unique. I am wondering when such a result can be extended to an infinite matrix. Say I have an infinite matrix $A$ in row ...
user3433489's user avatar
6 votes
1 answer
254 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 ...
Grešnik's user avatar
  • 1,802
2 votes
1 answer
501 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?
elipson_-1's user avatar
4 votes
2 answers
84 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 ...
Daniel P's user avatar
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