Hilbert matrix

Example

For $n \ge 1$, the Hilbert matrix $H_n$ is the $n \times n$ real matrix with entries $$(H_n)_{ij} = \frac{1}{i + j - 1} \qquad (1 \le i, j \le n).$$ For $n = 3$ it is $$H_3 = \begin{pmatrix} 1 & \tfrac12 & \tfrac13 \\ \tfrac12 & \tfrac13 & \tfrac14 \\ \tfrac13 & \tfrac14 & \tfrac15 \end{pmatrix}.$$

It is a Hankel matrix: the entry depends only on $i + j$, with $h_k = 1/(k-1)$.

It is a Cauchy matrix: with $x_i = i$ and $y_j = 1 - j$, the entry is $1/(x_i - y_j)$, and all $x_i - y_j = i + j - 1$ are nonzero.

It is a positive definite matrix. Indeed, $(H_n)_{ij} = \int_0^1 t^{i-1} t^{j-1} \, dt$, so $H_n$ is the Gram matrix of the functions $1, t, \dots, t^{n-1}$ for the inner product $\ip{f}{g} = \int_0^1 f(t) g(t) \, dt$. For $c \in \R^n$ and $p(t) = \sum_i c_i t^{i-1}$, this gives $c^T H_n c = \int_0^1 p(t)^2 \, dt$, which is positive when $c \ne 0$, because then $p$ is a nonzero polynomial.