Quadratic Mathematics | Part IB, 2003

Let SS be the set of all 2×22 \times 2 complex matrices AA which are hermitian, that is, A=AA^{*}=A, where A=AˉtA^{*}=\bar{A}^{t}.

(a) Show that SS is a real 4-dimensional vector space. Consider the real symmetric bilinear form bb on this space defined by

b(A,B)=12(tr(AB)tr(A)tr(B)).b(A, B)=\frac{1}{2}(\operatorname{tr}(A B)-\operatorname{tr}(A) \operatorname{tr}(B)) .

Prove that b(A,A)=detAb(A, A)=-\operatorname{det} A and b(A,I)=12tr(A)b(A, I)=-\frac{1}{2} \operatorname{tr}(A), where II denotes the identity matrix.

(b) Consider the three matrices

A1=(1001),A2=(0110) and A3=(0ii0)A_{1}=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right), \quad A_{2}=\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right) \quad \text { and } \quad A_{3}=\left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right)

Prove that the basis I,A1,A2,A3I, A_{1}, A_{2}, A_{3} of SS diagonalizes bb. Hence or otherwise find the rank and signature of bb.

(c) Let QQ be the set of all 2×22 \times 2 complex matrices CC which satisfy C+C=tr(C)IC+C^{*}=\operatorname{tr}(C) I. Show that QQ is a real 4-dimensional vector space. Given CQC \in Q, put

Φ(C)=1i2tr(C)I+iC.\Phi(C)=\frac{1-i}{2} \operatorname{tr}(C) I+i C .

Show that Φ\Phi takes values in SS and is a linear isomorphism between QQ and SS.

(d) Define a real symmetric bilinear form on QQ by setting c(C,D)=12tr(CD)c(C, D)=-\frac{1}{2} \operatorname{tr}(C D), C,DQC, D \in Q. Show that b(Φ(C),Φ(D))=c(C,D)b(\Phi(C), \Phi(D))=c(C, D) for all C,DQC, D \in Q. Find the rank and signature of the symmetric bilinear form cc defined on QQ.

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