| course |
course_year |
question_number |
tags |
title |
year |
Quadratic Mathematics |
IB |
66 |
IB |
2001 |
Quadratic Mathematics |
|
3.I.9B |
2001 |
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.]