Linear Algebra Done Right · Problem 7C.21

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · positive-operators

Problem

Suppose

n

is a positive integer. The

n

-by-

n

Hilbert matrix is the

n

-by-

n

matrix whose entry in row

j

, column

k

\frac{1}{j+k-1}

. Suppose

T \in \mathcal{L}(V)

is an operator whose matrix with respect to some orthonormal basis of

V

is the

n

-by-

n

Hilbert matrix. Prove that

T

is a positive invertible operator. Example: The 4-by-4 Hilbert matrix is

\begin{pmatrix} 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} \\ \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\ \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} \\ \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} \end{pmatrix}.