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I'm looking to see if there's a name for a particular type of matrix $M_{ij}=t_{\min(i,j)}$, ie.:

\begin{bmatrix} t_1 & t_1 & t_1 & t_1 \\ t_1 & t_2 & t_2 & t_2 & \cdots\\ t_1 & t_2 & t_3 & t_3 \\ t_1 & t_2 & t_3 & t_4 \\ & \vdots & & & \ddots \end{bmatrix}

Such a matrix has determinant $t(t_2-t_1)(t_3-t_2)(t_4-t_3)\cdots$ and its inverse is a very simple tridiagonal matrix. But it isn't a Vandermonde matrix or a Moore matrix. It looks like it's an alternant matrix, but that doesn't capture any of the interesting properties of the determinant or inverse. It seems like something with these special properties should be named or well-known somewhere.

This matrix came up in looking at a particular probabilistic process, where $P(x_1,t_1 ; x_2,t_2;\cdots)\propto \exp(-\frac{1}{2}\vec{x}^T M^{-1}\vec{x})$ (hence the significance of the simple tridiagonal structure of $M^{-1}$).

user1551
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David
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  • This is an example of a GCD matrix in the wider sense of this word, i.e., of an $n\times n$-matrix whose $\left(i,j\right)$-th element is $a_{u_i \wedge u_j}$, where $u_1, u_2, \ldots, u_n$ are some $n$ elements of a lattice $L$, and $a_k$ is a number for each $k \in L$. There is noticeable literature about these matrices, specifically when $L$ is the divisor lattice of the integers (see, e.g., https://mathoverflow.net/questions/262153/the-gcd-matrix-generalizing-a-result-of-smith ), and I'm pretty sure at least some of it considers the general case, but I can't find it right now. – darij grinberg Oct 22 '18 at 22:31
  • Ah, these slides by Bruce Sagan seem friendly: http://users.math.msu.edu/users/sagan/Slides/mfp5.pdf – darij grinberg Oct 22 '18 at 22:34
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    Aha -- a meet matrix. – darij grinberg Oct 22 '18 at 22:39

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