(I had this question in mind for longer time, but it is just triggered now by some comments at that recent question in mse)
(Background) I was looking at properties of the Pascal-matrix:
$\qquad \qquad P_{n=5}=\small \begin{bmatrix} 1 & 1 & 1 & 1 & 1 \\ . & 1 & 2 & 3 & 4 \\ . & . & 1 & 3 & 6 \\ . & . & . & 1 & 4 \\ . & . & . & . & 1 \end{bmatrix} $ where the index $n$ indicates the size of the matrix.
This matrix, for any size n x n is not diagonalizable. However, in the case of infinite size we can have $$ \lim_{ n\to \infty} P_n \cdot M_n = M_n \cdot D_n $$ or simply $$ P \cdot M = M \cdot D \qquad \qquad \text{matrices of infinite size}$$(D diagonal) with some matrix $M$ (see below) but where we cannot write $ P = M \cdot D \cdot M^{-1}$ because $M^{-1}$ does not exist for that case of infinite size.
(M has the form $M :m_{r,c} = { c^r \over r!} \qquad \small \text{where } r,c \ge 0 $)
Question: Can I thus write in an article: "P when assumed of infinite size is diagonalizable" ?
Or would there be another (canonical) expression/terminology for this?
Appendix 1
The top left 5x5 segment of the infinite matrix $M$ is $ \displaystyle M = \Tiny \begin{bmatrix} 1 & 1 & 1 & 1 & 1 & \cdots \\ 0 & 1 & 2 & 3 & 4 & \cdots \\ 0 & 1/2 & 2 & 9/2 & 8 & \cdots \\ 0 & 1/6 & 4/3 & 9/2 & 32/3 & \cdots \\ 0 & 1/24 & 2/3 & 27/8 & 32/3 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{bmatrix}$
The infinite matrix $D$ is $ \displaystyle diag([1,e,e^2,e^3,e^4,...])$
Appendix 2
(If we introduce a symbol $ \# $ for a formal notation of matrix-multiplication, but where the evaluation is not allowed due to divergence/singularity in the dot-products we can write a nearly perfect diagonalization formula with all triangular invertible matrices $$ P = (S_2 \cdot P) \cdot D \cdot (P^{-1} \# S_1) \\ P^m = (S_2 \cdot P) \cdot D^m \cdot (P^{-1} \# S_1)$$ where the matrices $S_2$ and $S_1$ are the arrays of Stirling numbers second resp first kind, scaled by factorials and $S_1 = S_2^{-1}$ also in the case of infinite size (as mentioned for instance in Abramowicz&Stegun))