Quadratic Mathematics | Part IB, 2001

Let AA be the Hermitian matrix

(1i2ii3i2ii5)\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 DD with rational entries, and an invertible (complex) matrix TT such that TDT=AT^{*} D T=A, where TT^{*} here denotes the conjugated transpose of TT.

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

[You may assume that if a monic polynomial t3+a2t2+a1t+a0t^{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.]

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