Hilbert matrices
For a positive integer \(n\), prove that the Hilbert matrix
\[H=\left(\frac{1}{i+j-1}\right)_{i,j=1}^{n}\]is nonsingular.
The solution uses definite integrals.
Hint
Write each entry as the integral of a power of \(t\).
Solution
The entries are inner products of monomials:
\[H_{ij}=\int_0^1 t^{i-1}t^{j-1}\,dt.\]For any nonzero real column vector \(v=(v_1,\ldots,v_n)^\mathsf{T}\),
\[v^\mathsf{T}Hv =\int_0^1\left(\sum_{i=1}^{n}v_it^{i-1}\right)^2\,dt>0.\]The polynomial inside the square is nonzero. It has only finitely many roots and is continuous, so its square is positive on some interval in \((0,1)\). This justifies the strict inequality.
If \(Hv=0\), then \(v^\mathsf{T}Hv=0\), forcing \(v=0\). Hence \(H\) is nonsingular.