Paper 1, Section I, 2 B2 \mathrm{~B}

Vectors and Matrices | Part IA, 2009

Define the Hermitian conjugate AA^{\dagger} of an n×nn \times n complex matrix AA. State the conditions (i) for AA to be Hermitian (ii) for AA to be unitary.

In the following, A,B,CA, B, C and DD are n×nn \times n complex matrices and x\mathbf{x} is a complex nn-vector. A matrix NN is defined to be normal if NN=NNN^{\dagger} N=N N^{\dagger}.

(a) Let AA be nonsingular. Show that B=A1AB=A^{-1} A^{\dagger} is unitary if and only if AA is normal.

(b) Let CC be normal. Show that Cx=0|C \mathbf{x}|=0 if and only if Cx=0\left|C^{\dagger} \mathbf{x}\right|=0.

(c) Let DD be normal. Deduce from (b) that if ee is an eigenvector of DD with eigenvalue λ\lambda then ee is also an eigenvector of DD^{\dagger} and find the corresponding eigenvalue.

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