Have symmetric matrices for which the $(i,j)$ entry is $\min(i,j)$ earned a name?

Question

I am interested in symmetric matrices where the $(i,j)$ entry is $\min(i,j)$, i.e.,

$$\begin{pmatrix} 1 & 1 & 1 & \dots & 1 & 1 \\ 1 & 2 & 2 & \dots & 2 & 2 \\ 1 & 2 & 3 & \dots & 3 & 3 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ 1 & 2 & 3 & \dots & n-1 & n-1 \\ 1 & 2 & 3 & \dots & n-1 & n \end{pmatrix}$$

Note that these symmetric matrices are related to Lehmer matrices. Have these symmetric matrices earned a name?

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Motivation

These matrices are interesting because they are the Gramians of binary triangular matrices, e.g.,

$$ \begin{bmatrix}1 & 1 & 1 & 1 & 1\\ 0 & 1 & 1 & 1 & 1\\ 0 & 0 & 1 & 1 & 1\\ 0 & 0 & 0 & 1 & 1\\ 0 & 0 & 0 & 0 & 1 \end{bmatrix}^\top \begin{bmatrix}1 & 1 & 1 & 1 & 1\\ 0 & 1 & 1 & 1 & 1\\ 0 & 0 & 1 & 1 & 1\\ 0 & 0 & 0 & 1 & 1\\ 0 & 0 & 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 & 1 & 1\\ 1 & 2 & 2 & 2 & 2\\ 1 & 2 & 3 & 3 & 3\\ 1 & 2 & 3 & 4 & 4\\ 1 & 2 & 3 & 4 & 5\end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 & 0\\ 1 & 1 & 0 & 0 & 0\\ 1 & 1 & 1 & 0 & 0\\ 1 & 1 & 1 & 1 & 0\\ 1 & 1 & 1 & 1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0 & 0\\ 1 & 1 & 0 & 0 & 0\\ 1 & 1 & 1 & 0 & 0\\ 1 & 1 & 1 & 1 & 0\\ 1 & 1 & 1 & 1 & 1 \end{bmatrix}^\top$$

Thus, the determinant of these symmetric matrices is $1$.

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Some findings

• In 2002, Mario Catalani studied these matrices

• In 2006, Rajendra Bhatia called these $\min$ matrices

• In 2016, Mika Mattila & Pentti Haukkanen called the $n \times n$ matrix of this form the MIN matrix of the set $\{1,2,\dots,n\}$

• In 2023, Darij Grinberg studied a generalization of these matrices.

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Related

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• Find rank of $A$ with entries $a_{ij}=\min\{i,j\} , 1\leq i,j \leq n $.

• Inverse of a symmetric tridiagonal filter matrix

• Properties of square array of numbers min (i,j)

• Name of a particular matrix with $M_{ij}=t_{\min(i,j)}$?

• Does this type of matrix with the same diagonals have a name?

Answer 1

This is not to suggest a terminology, but to inform a relationship of the matrix to (algebraic) geometry after the comment. Thank you for your interest.

As one of the links shows, the inverse of our matrix is the negative of $$G=\begin{pmatrix} -2 & 1 \\ 1 & -2 & 1 \\ & 1 & -2 & 1 \\ &&&\ddots\\ &&& 1 & -2 &1\\ &&&& 1 & -1 \end{pmatrix}.$$ This matrix is classically known to arise in projective geometry as follows. Recall that for a 2-dimensional complex manifold $Z$ and its point $z\in Z$, we can perform blowing up at $z$: it creates a new manifold $Z_1$ with a map $\pi_1\colon Z_1\rightarrow Z$, which has the property that $Z_1-\pi_1^{-1}(z)\simeq Z-\{z\}$ via $\pi_1$ (bijective, isomorphic) and $E_1:=\pi_1^{-1}(z)\simeq \mathbb{P}^1$ is isomorphic to a projective line. In other words, $Z_1$ is the manifold which is identical to $Z$ except that the center $z\in Z$ is replaced by an exceptional curve $E_1\subset Z_1$.

Now, starting from $Z=\mathbb{C}^2$, we repeat blowings up $n$ times and get maps $$X=Z_n \stackrel{\pi_n}{\rightarrow} Z_{n-1}\stackrel{\pi_{n-1}}{\rightarrow} \cdots \stackrel{\pi_2}{\rightarrow} Z_1\stackrel{\pi_1}{\rightarrow} Z.$$ The center of $\pi_i$ is chosen to be a general (i.e., not special) point of the previous exceptional curve $E_{i-1}\subset Z_{i-1}$. (In fact, the choice is not very important.) The resulting $X$ has $n$ exceptional curves, which form a basis of the middle dimensional homology group $H_2(X,\mathbb{Z})$.

As a general theory via the Poincar'{e} duality, we have the intersection pairing $$H_2(X,\mathbb{Z})\times H_2(X,\mathbb{Z})\rightarrow \mathbb{Z}$$ which gives a symmetric bilinear form on $H_2(X,\mathbb{Z})$. Some computations show that the form values at the basis are $$(E_i,E_j)=\delta_{1,|i-j|},\ (E_n^2)=-1,\ (E_k^2)=-2\ (1\leq i,j\leq n, 1\leq k\leq n-1),$$ which can be seen also as a consequence of basic theorems in intersection theory. Therefore, the Gram matrix with respect to the basis $E_1,\dots,E_n$ coincides with $G$ above.

Finally, we note that some of the properties of the matrix are closely related to theorems in geometry: the intersection pairing is unimodular by Poincar'{e} duality, which gives that the determinant is $\pm 1$. Also, the (negative-) definiteness of the matrix can be geometrically proved from the Hodge index theorem.