16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 9.3.12

Operators on Inner Product Spaces

Mathematics — Linear Algebra · hilbert-matrix-positive

Problem

Let

n

be a positive integer and

A

the

n \times n

matrix

A = \begin{bmatrix} 1 & \dfrac{1}{2} & \dfrac{1}{3} & \cdots & \dfrac{1}{n} \\[6pt] \dfrac{1}{2} & \dfrac{1}{3} & \dfrac{1}{4} & \cdots & \dfrac{1}{n+1} \\[6pt] \vdots & \vdots & \vdots & & \vdots \\[6pt] \dfrac{1}{n} & \dfrac{1}{n+1} & \dfrac{1}{n+2} & \cdots & \dfrac{1}{2n-1} \end{bmatrix}.

Prove that

A

is positive.