Question

Express A as the sum of a hermitian and a skew- hermitian matrix, where
$\sf{A = \: \bigg[ \begin{matrix}2 + 3i& 7\: \\ 1 - i& 2i \end{matrix} \bigg],i = \sqrt{ - 1} }$
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Answer 1

$As we have , \text { Hermitian natuin of } A \text { is } A^{H} \text {; }$

$A^{H}=\overline{\left[A^{\top}\right]}\\=\left[\begin{array}{cc}2+3 i & 1-i \\7 & 2 i\end{array}\right]\\A^{H}=\left[\begin{array}{cc}2-3 i & 1+i \\7 & -2 i\end{array}\right]\\P=\frac{1}{2}\left(A+A^{n}\right)$

$=\frac{1}{2}\left\{\left[\begin{array}{cc}2+3 i & 7 i \\1-i & 2 i\end{array}\right]\right.+\left.+\left[\begin{array}{cc}2-3 i & 1+i \\7 & -2 i\end{array}\right]\right\}$

=$\frac{1}{2}\left\{\left[\begin{array}{ll}4 & 8+2 \\8-i & 0\end{array}\right]\right\}$

=$=\left[\begin{array}{cc}2 & 4+i 12 \\4-i / 2 & 0\end{array}\right]$

=$P=\left[\begin{array}{cc}2 & 4+i / 2 \\4-i / 2 & 0\end{array}\right]$

=$p^{H}=\left[\overline{p^{\top}}\right]\\=\left[\begin{array}{cc}2 & 4-i / 2 \\4+i & 4-2 / 2 \\4+L & 0\end{array}\right]$

=$=\left[\begin{array}{cc}2 & 4+i / 2 \\4-i / 2 & 0\end{array}\right]\\=P^{H}=P\\\\P \text { is Hermitian matrix }\\\\\text { Let } q=\frac{1}{2}\left(A-A^{H}\right) \text {; }\\q=\frac{1}{2}\left\{\left[\begin{array}{cc}2+3 i & 7 \\1-i & 2 i\end{array}\right]\right.-\left[\begin{array}{cc}2+3 i & 1+i \\7 & -2 i\end{array}\right]$

=$q=\frac{1}{2}\left[\begin{array}{cc}6 i & 6-i \\-6-i & 4 i\end{array}\right]\\=\left[\begin{array}{cc}3 \dot{L} & 3-i_{12} \\-3-i / 2 & 2 i\end{array}\right]\\q^{H}=\left[\bar{q}^{-}\right]\\=\left[\begin{array}{cc}3 i-3-\frac{i}{2} \\3-\frac{i}{2} & 2\\\end{array}\right]\\\\=-\left[\begin{array}{cc}3 i & 3-i / 2 \\\\\\-3-i / 2 & 2 i\end{array}\right]\\\begin{array}{l}q^{4}=-q \\q=-q^{H}\end{array}$

$\therefore \quad q \text { is skew Hermitian matiix. }\\\text { Inus, } A=p+q=\frac{1}{2}\left(A+A^{H}\right)+\frac{1}{2}\left(A-A^{H}\right)\\\\=A+\frac{1}{2}\left(A^{H}-A^{H}\right)\\\\=A\\A=\frac{1}{2}\left(A+A^{H}\right)+\frac{1}{2}\left(A-A^{H}\right)$

Hence , above equation show

A = Hermitian + skew-Hermitian

Answer 2

Answer:

this is the answer I think sososo