# 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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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### 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 ...