I'm looking at this matrix:
$$
\begin{pmatrix}
1 & 1/2 & 1/8 & 1/48 & 1/384 & \dots \\
0 & 1/2 & 1/4 & 1/16 & 1/96 & \dots \\
0 & 0 & 1/8 & 1/16 & 1/64 & \dots \\
0 & 0 & 0 & 1/48 & 1/96 & \dots \\
0 & 0 & 0 & 0 & 1/384 & \dots \\
\vdots & \vdots & \vdots & \vdots & \vdots & \ddots
\end{pmatrix}
$$
The first row contains the reciprocals of the double factorials
$$
2, \qquad 2 \cdot 4, \qquad 2 \cdot 4 \cdot 6, \qquad 2 \cdot 4 \cdot 6 \cdot 8, \qquad \dots
$$
Each row is a shift of a scalar multiple of the first row, and the scalar multiple is in each case itself a reciprocal of a double factorial, so that the main diagonal is the same as the first row. A consequence is that each column is proportional to the corresponding row of Pascal's triangle. E.g. the last column shown is proportional to
$$
1, 4, 6, 4, 1.
$$
This matrix is the matrix of coefficients in the "inversion formulas" section of this rant that I wrote.
I found the first three eigenvectors:
$$
\begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ \vdots \end{pmatrix},
\begin{pmatrix} 1 \\ -1 \\ 0 \\ 0 \\ 0 \\ \vdots \end{pmatrix},
\begin{pmatrix} 5 \\ -14 \\ 21 \\ 0 \\ 0 \\ \vdots \end{pmatrix}
$$
Meni Rosenfeld pushed this through some software and found that up to the 40th eigenvalue, the signs of the components of the eigenvectors alternate.
Can anything of interest be said about the eigenvectors?
Can anything of interest be said about this matrix?
Best Answer
As far as I understand your construction, your matrix is $$ \operatorname{diag}\biggl(1,\frac12,\frac18,\dots,\frac1{2^nn!},\dots\biggr) \cdot\exp\begin{pmatrix} 0 & \frac12 & 0 & 0 & 0 & \dots \cr 0 & 0 & \frac12 & 0 & 0 & \dots \cr 0 & 0 & 0 & \frac12 & 0 & \dots \cr 0 & 0 & 0 & 0 & \frac12 & \dots \cr \dots & \dots & \dots & \dots & \dots & \dots \end{pmatrix}, $$ a diagonal matrix times the exponential of a nilpotent matrix. In your question you discuss some properties of truncations of your infinite matrix, finite $n\times n$ matrices. This corresponds to the truncations of the above diagonal and nilpotent matrices. I've never seen such matrices "in work" but this one could be a nice example of understanding the alteration property of entries of its eigenvectors. In view of the other response, this could be a good point of generalising the previous results in this area.