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Logarithmic Differentiation

Expressing ...

Question

Expressing $A$ as the sum of Hermitian and a Skew-Hermitian matrix, where,
$A = ⎡ ⎢ ⎣ \begin{matrix} 2 + 3 i & 2 & 5 - 3 - i & 7 & 3 - i 3 - 2 i & i & 2 + i \end{matrix} ⎤ ⎥ ⎦$
be $A = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 2 & - \frac{1}{2} + \frac{i}{2} & m + k i - \frac{1}{2} - \frac{i}{2} & 7 & \frac{3}{2} - 1 4 - i & \frac{3}{2} + 1 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ + ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 3 i & \frac{5}{2} - \frac{i}{2} & n + p i - \frac{5}{2} - \frac{i}{2} & 0 & \frac{3}{2} - 1 - i & - \frac{3}{2} & i \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$ .Find $m + k + n + p$ ?

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Solution

$A = ⎡ ⎢ ⎣ \begin{matrix} 2 + 3 i & 2 & 5 - 3 - i & 7 & 3 - i 3 - 2 i & i & 2 + i \end{matrix} ⎤ ⎥ ⎦$
$∴ A^{'} = ⎡ ⎢ ⎣ \begin{matrix} 2 + 3 i & - 3 - i & 3 - 2 i 2 & 7 & i 5 & 3 - i & 2 + i \end{matrix} ⎤ ⎥ ⎦$
$¯ ¯¯¯¯¯¯¯¯ ¯ (A^{'}) = ⎡ ⎢ ⎣ \begin{matrix} 2 - 3 i & - 3 + i & 3 + 2 i 2 & 7 & - i 5 & 3 + i & 2 - i \end{matrix} ⎤ ⎥ ⎦$
Or $A^{⊖} = ⎡ ⎢ ⎣ \begin{matrix} 2 - 3 i & - 3 + i & 3 + 2 i 2 & 7 & - i 5 & 3 + i & 2 - i \end{matrix} ⎤ ⎥ ⎦$
$∴ A + A^{⊖} = ⎡ ⎢ ⎣ \begin{matrix} 2 + 3 i & 2 & 5 - 3 - i & 7 & 3 - i 3 - 2 i & i & 2 + i \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 2 - 3 i & - 3 + i & 3 + 2 i 2 & 7 & - i 5 & 3 + i & 2 - i \end{matrix} ⎤ ⎥ ⎦$
$= ⎡ ⎢ ⎣ \begin{matrix} 4 & - 1 + i & 8 + 2 i - 1 - i & 14 & 3 - 2 i 8 - 2 i & 3 + 2 i & 4 \end{matrix} ⎤ ⎥ ⎦ . . . (1)$
And $A - A^{⊖} = ⎡ ⎢ ⎣ \begin{matrix} 2 + 3 i & 2 & 5 - 3 - i & 7 & 3 - i 3 - 2 i & i & 2 + i \end{matrix} ⎤ ⎥ ⎦ - ⎡ ⎢ ⎣ \begin{matrix} 2 - 3 i & - 3 + i & 3 + 2 i 2 & 7 & - i 5 & 3 + i & 2 - i \end{matrix} ⎤ ⎥ ⎦$
$= ⎡ ⎢ ⎣ \begin{matrix} 6 i & 5 - i & 2 - 2 i - 5 - i & 0 & 3 - 2 - 2 i & - 3 & 2 i \end{matrix} ⎤ ⎥ ⎦ . . . (2)$
Adding $(1)$ and $(2)$ , we get
$2 A = ⎡ ⎢ ⎣ \begin{matrix} 4 & - 1 + i & 8 + 2 i - 1 - i & 14 & 3 - 2 i 8 - 2 i & 3 + 2 i & 4 \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 6 i & 5 - i & 2 - 2 i - 5 - i & 0 & 3 - 2 - 2 i & - 3 & 2 i \end{matrix} ⎤ ⎥ ⎦$
Hence,
$A = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 2 & - \frac{1}{2} + \frac{i}{2} & 8 + 2 i - \frac{1}{2} - \frac{i}{2} & 7 & \frac{3}{2} - 1 4 - i & \frac{3}{2} + 1 & 2 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ + ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 3 i & \frac{5}{2} - \frac{i}{2} & 1 - i - \frac{5}{2} - \frac{i}{2} & 0 & \frac{3}{2} - 1 - i & - \frac{3}{2} & i \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$
$\Rightarrow m = 8, k = 2, n = 1$ and $p = - 1$
$∴ m + k + n + p = 10$

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