Eigenvalues for structured matrix ($L^TL$ for constant lower triangular matrix)

Asked Oct 02 '21 at 23:33

Active Oct 02 '21 at 23:33

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Consider the $d \times d$ lower triangular matrix $L$ with entries $L_{ij} = 1$, $j \leq i$, and $L_{ij} = 0$ otherwise.

I am interested in computing the eigenvalues of $L^T D L$ with $D$ a diagonal matrix.

Is this easy to do in closed form? Even for the case $D = I$, I would be interested in help with this.

Also, do these type of matrices have a name? They seem to fall under the category of "arrow" structure.

asked Oct 02 '21 at 23:33

Drew Brady

Look at https://math.stackexchange.com/questions/264969/proof-that-eigenvalues-are-the-diagonal-entries-of-the-upper-triangular-matrix-i – Rishi Sonthalia Oct 03 '21 at 00:00
@RishiSonthalia The question you linked is about the eigenvalues of $L$. This question is about the eigenvalues of $L^TL$. – Ben Grossmann Oct 03 '21 at 02:20
@Drew This post and this post might be relevant – Ben Grossmann Oct 03 '21 at 02:21
@Drew If I remember correctly, the inverse of either $L^TL$ or $LL^T$ is tridiagonal and Toeplitz, which would lead to a nice closed form for the eigenvalues of $L^TL$. – Ben Grossmann Oct 03 '21 at 02:24
This is pretty useful, thanks! I was aware that tridiagonal/Toeplitz had closed form eigenvalues. Is there a characterization of the inverse of such matrices? – Drew Brady Oct 04 '21 at 04:45

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