Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2002 / Paper 1 / 5 / b

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- Solution b

Solution

C

is a unitary matrix,

C^{- 1} = C^{†}

. Let

M = C^{†} A C

. Its conjugate transpose is

M^{†} = C^{†} A^{†} C = M

, so

M

and every positive integer power

M^{n}

are Hermitian matrices and have real eigenvalues.

More explicitly, if

A v_{j} = a_{j} v_{j}

with

a_{j}

real, then

C^{- 1} v_{j}

form an eigenbasis and

M^{n} (C^{- 1} v_{j}) = C^{- 1} A^{n} v_{j} = a_{j}^{n} (C^{- 1} v_{j}) .

Therefore

spec ((C^{- 1} A C)^{n}) = {a_{j}^{n} : j = 1, \dots, n_{A}} \subset R,

where

n_{A}

denotes the matrix size, to distinguish it from the power

n

.

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