Q.1 (d) Express the matrix \(A=\begin{bmatrix}2+3i&2&3i\\-2i&0&1+2i\\4&2+5i&-i\end{bmatrix}\) as the sum of a Hermitian and a skew-Hermitian matrix.
Best answer
\(A=P+Q\) with \(P=\frac12(A+A^\theta)\) (Hermitian) and \(Q=\frac12(A-A^\theta)\) (skew-Hermitian), where \(A^\theta=(\bar A)^T\).
\(A^\theta=\begin{bmatrix}2-3i&2i&4\\2&0&2-5i\\-3i&1-2i&i\end{bmatrix}\)
\(P=\frac12\begin{bmatrix}4&2+2i&4+3i\\2-2i&0&3-3i\\4-3i&3+3i&0\end{bmatrix}=\begin{bmatrix}2&1+i&2+\frac32i\\1-i&0&\frac32-\frac32i\\2-\frac32i&\frac32+\frac32i&0\end{bmatrix}\) — real diagonal and \(p_{ji}=\bar p_{ij}\), so \(P^\theta=P\).
\(Q=\frac12\begin{bmatrix}6i&2-2i&-4+3i\\-2-2i&0&-1+7i\\4+3i&1+7i&-2i\end{bmatrix}=\begin{bmatrix}3i&1-i&-2+\frac32i\\-1-i&0&-\frac12+\frac72i\\2+\frac32i&\frac12+\frac72i&-i\end{bmatrix}\) — purely imaginary diagonal and \(q_{ji}=-\bar q_{ij}\), so \(Q^\theta=-Q\).
\(P+Q=A\) (e.g. \(2+3i\), \((1+i)+(1-i)=2\), \((2+1.5i)+(-2+1.5i)=3i\)).
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