3.I.9B

Let $A$ be the Hermitian matrix

$\left(\begin{array}{rrr} 1 & i & 2 i \\ -i & 3 & -i \\ -2 i & i & 5 \end{array}\right)$

Explaining carefully the method you use, find a diagonal matrix $D$ with rational entries, and an invertible (complex) matrix $T$ such that $T^{*} D T=A$, where $T^{*}$ here denotes the conjugated transpose of $T$.

Explain briefly why we cannot find $T, D$ as above with $T$ unitary.

[You may assume that if a monic polynomial $t^{3}+a_{2} t^{2}+a_{1} t+a_{0}$ with integer coefficients has all its roots rational, then all its roots are in fact integers.]

*Typos? Please submit corrections to this page on GitHub.*