Exercise 10.A.20

Proof. As explained in the solution to Exercise 18, the right side of the inequality is equal to $\mathrm{trace}<!--\; FUNCTION\; APPLICATION\; -->(T^{\text{∗}}T)$. Let $f_1,\dots ,f_n$ be an orthonormal basis of $V$ with respect to which the matrix of $T$ is upper triangular (6.37 assures the existence of this basis). Then the eigenvalues of $T$ appear on the diagonal of this matrix. Because $M(T^{\text{∗}})$ is the conjugate transpose of $M(T)$, where this matrices are with respect to the basis $f_1,\dots ,f_n$, a moment’s thought shows that

The right side of the inequality above equals $\mathrm{trace}<!--\; FUNCTION\; APPLICATION\; -->(T^{\text{∗}}T)$, which yields the desired result. □