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 infinite matrix A. (So that we can easily analyze that matrix with computer)
(If "Best match" is ambiguous, let's start by "in terms of multiplication".)
If not every infinite matrix can be cut-off(Converting to finite matrix is so useless that it cannot represent the original matrix at all), than what conditions should the infinite matrix fit?
 A: This kinda depends on the properties of the infinite matrix / linear operator (after all, $\bf A$ is nothing but the representation matrix $(\langle e_i,Ae_j\rangle)_{i,j\in\mathbb N}$ of your operator $A$ with respect to an orthonormal basis $(e_n)_{n\in\mathbb N}$ of your Hilbert space $\mathcal H$) you are considering. For this, let $\Pi_n$ is the orthogonal projection $\sum_{j=1}^n\langle e_n,\cdot\rangle e_n$ onto the first $n$ elements of your orthonormal basis $e_1,\ldots,e_n$, so $\Pi_n {\bf A}\Pi_n$ is the upper left $n\times n$-block of $\bf A$ (and the rest of $\bf A$ is set to zero).
Now, for example: if $\bf A$ is compact, then $\Pi_n{\bf A}\Pi_n$ approximates ${\bf A}$ in the operator norm, e.g. $\|{\bf A}-\Pi_n{\bf A}\Pi_n\|\to 0$ as $n\to\infty$. Even stronger, if $\bf A$ is a Schatten class operator with $p\in [1,\infty)$, then the above approximation holds in the $p$-norm so $\|{\bf A}-\Pi_n{\bf A}\Pi_n\|_p\to 0$ as $n\to\infty$. Furthermore, these statements hold for all orthonormal bases of $A$ so you can choose any represenetation matrix of your operator $\bf A$ (e.g. $U{\bf A}U^\dagger$ for any unitary $U\in\mathcal B(\mathcal H)$ ).
Then again, your exact question was kinda vague to me so I'm not sure if this is what you were looking for - if I missed your point, I'm sorry.
