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 already but I would like to confirm that I can really use it in revisions of my web-essays.
Consider the (upper triangular) infinite "Pascal"/"binomial"-matrix P with top-left element as
$$\small \begin{bmatrix}
1 & 1 & 1 & 1 & 1 & 1 \\
. & 1 & 2 & 3 & 4 & 5 \\
. & . & 1 & 3 & 6 & 10 \\
. & . & . & 1 & 4 & 10 \\
. & . & . & . & 1 & 5 \\
. & . & . & . & . & 1
\end{bmatrix}$$
Rightmultiplying it with the columnvector $E_1 = [1,1/1!,1/2!,1/3!, \cdots]$ gives
$$ P \cdot E_1 = e \cdot E_1
$$
which has the form of an eigenvector equation as known from the cases with matrices of finite size. However, using $P$ and $E_1$ truncated to finite size $P^\star$ this would never be correct since $P^\star$ has no diagonalization.
Back to infinite size: in general with some columnvector $E(x)=[1, x^1, x^2/2!, x^3/3!, \cdots]$ we have
$$ P \cdot E(x) = e^x \cdot E(x)
$$
thus for each $x$ we had that $P$ has $E(x)$ as eigenvector to eigenvalue $e^x$.
Now what I'm discussing in a couple of essays are a second type of infinite matrices, namely the concatenation of vectors $E_n=E(n)$ to a matrix $$EZ=[E_0,E_1,E_2,E_3,...]$$
and following the example I could write
$$ P \cdot EZ = EZ \cdot \,^dV(e) \\\qquad \qquad \qquad \small{\text{ where $\,^dV(e)$ = diagonal}([1,e,e^2,e^3...])}
$$
which has again the form of an eigenmatrix-decomposition.
I always tended to say, that
- "EZ is an eigenmatrix of P" , or that
- "P of infinite size has a diagonalization"
and used this at several places in my manuscripts.
But because for the case of finite size P$\,^\star$ has no diagonalization (it has only a Jordan-form), I feel it might be too sloppy to formulate this as an Eigenmatrix-relation or even as "diagonalization of P" (the latter is even more problematic since the matrix EZ has no inverse/reciprocal and we cannot write $\text{P}=\text{EZ} \cdot \,^dV(e) \cdot \text{EZ}^{-1}$).
Q: How could I correctly express that relation, even in a informal context? Can I still apply the terms "eigenmatrix" and "diagonalization"?