Paper 1, Section II, C

Vectors and Matrices | Part IA, 2014

Let AA and BB be complex n×nn \times n matrices.

(i) The commutator of AA and BB is defined to be

[A,B]ABBA[A, B] \equiv A B-B A

Show that [A,A]=0;[A,B]=[B,A];[A, A]=0 ;[A, B]=-[B, A] ; and [A,λB]=λ[A,B][A, \lambda B]=\lambda[A, B] for λC\lambda \in \mathbb{C}. Show further that the trace of [A,B][A, B] vanishes.

(ii) A skew-Hermitian matrix SS is one which satisfies S=SS^{\dagger}=-S, where \dagger denotes the Hermitian conjugate. Show that if AA and BB are skew-Hermitian then so is [A,B][A, B].

(iii) Let M\mathcal{M} be the linear map from R3\mathbb{R}^{3} to the set of 2×22 \times 2 complex matrices given by

M(xyz)=xM1+yM2+zM3\mathcal{M}\left(\begin{array}{l} x \\ y \\ z \end{array}\right)=x M_{1}+y M_{2}+z M_{3}


M1=12(i00i),M2=12(0110),M3=12(0ii0)M_{1}=\frac{1}{2}\left(\begin{array}{cc} i & 0 \\ 0 & -i \end{array}\right), \quad M_{2}=\frac{1}{2}\left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right), \quad M_{3}=\frac{1}{2}\left(\begin{array}{cc} 0 & i \\ i & 0 \end{array}\right) \text {. }

Prove that for any aR3,M(a)\mathbf{a} \in \mathbb{R}^{3}, \mathcal{M}(\mathbf{a}) is traceless and skew-Hermitian. By considering pairs such as [M1,M2]\left[M_{1}, M_{2}\right], or otherwise, show that for a,bR3\mathbf{a}, \mathbf{b} \in \mathbb{R}^{3},

M(a×b)=[M(a),M(b)]\mathcal{M}(\mathbf{a} \times \mathbf{b})=[\mathcal{M}(\mathbf{a}), \mathcal{M}(\mathbf{b})]

(iv) Using the result of part (iii), or otherwise, prove that if CC is a traceless skewHermitian 2×22 \times 2 matrix then there exist matrices A,BA, B such that C=[A,B]C=[A, B]. [You may use geometrical properties of vectors in R3\mathbb{R}^{3} without proof.]

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