Hilbert matrix ($H_{ij}=\frac{1}{i+j-1},1\le i,j\le n$)

A real symmetric matrix whose entries are the moments $H_{ij}={\int }_0^1{x}^{i-1}{x}^{j-1}dx$. Thus it is the Gram matrix of the monomials in the real L2 inner product on $[0,1]$. For a nonzero real coefficient vector $v$, $v^{T}Hv={\int }_0^1({\sum }_i{v}_i{x}^{i-1}{)}^{2}dx>0$, so every finite Hilbert matrix is positive-definite. The moment description connects these matrices with polynomial approximation and with the shifted Hilbert quadratic-form bound.

For real $a_r$, put $h(z)={\sum }_{r=1}^n{a}_r{z}^r$. The quadratic form equals ${\int }_0^{1}h(x{)}^{2}dx$, proving its nonnegativity. Apply the centred logarithmic keyhole identity bound to the entire function $h^2$. Since $|h(e^{i\theta }{)}^2|=|h(e^{i\theta }){|}^2$, orthogonality of the finite Fourier sum gives ${\int }_0^{2\pi }|h(e^{i\theta }){|}^{2}d\theta =2\pi {\sum }_r{a}_r^2$. Multiplication by the contour estimate's factor $1/2$ proves the upper bound. The matrix $(1/(r+s+1))$ is the principal submatrix of a Hilbert matrix obtained by deleting its first row and column.