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Symmetric Matrix

Express the m...

Question

Express the matrix $B = ⎡ ⎢ ⎣ \begin{matrix} 2 & - 2 & - 4 - 1 & 3 & 4 1 & - 2 & - 3 \end{matrix} ⎤ ⎥ ⎦$ as the sum of a symmetric and a skew symmetric matrix.

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Solution

Given: $B = ⎡ ⎢ ⎣ \begin{matrix} 2 & - 2 & - 4 - 1 & 3 & 4 1 & - 2 & - 3 \end{matrix} ⎤ ⎥ ⎦$
$B^{'} = ⎡ ⎢ ⎣ \begin{matrix} 2 & - 1 & 1 - 2 & 3 & - 2 - 4 & 4 & - 3 \end{matrix} ⎤ ⎥ ⎦$

Let $P = \frac{1}{2} (B + B^{'})$
$\Rightarrow P = \frac{1}{2} ⎛ ⎜ ⎝ ⎡ ⎢ ⎣ \begin{matrix} 2 & - 2 & - 4 - 1 & 3 & 4 1 & - 2 & - 3 \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 2 & - 1 & 1 - 2 & 3 & - 2 - 4 & 4 & - 3 \end{matrix} ⎤ ⎥ ⎦ ⎞ ⎟ ⎠$
$\Rightarrow P = \frac{1}{2} ⎡ ⎢ ⎣ \begin{matrix} 4 & - 3 & - 3 - 3 & 6 & 2 - 3 & 2 & - 6 \end{matrix} ⎤ ⎥ ⎦$
$\Rightarrow P = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 2 & \frac{- 3}{2} & \frac{- 3}{2} \frac{- 3}{2} & 3 & 1 \frac{- 3}{2} & 1 & - 3 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$
$P^{'} = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 2 & \frac{- 3}{2} & \frac{- 3}{2} \frac{- 3}{2} & 3 & 1 \frac{- 3}{2} & 1 & - 3 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = P$
$∴ P^{'} = P$
$∴ P$ is a symmetric matrix.

Let $Q = \frac{1}{2} (B - B^{'})$
$\Rightarrow Q = \frac{1}{2} ⎛ ⎜ ⎝ ⎡ ⎢ ⎣ \begin{matrix} 2 & - 2 & - 4 - 1 & 3 & 4 1 & - 2 & - 3 \end{matrix} ⎤ ⎥ ⎦ - ⎡ ⎢ ⎣ \begin{matrix} 2 & - 1 & 1 - 2 & 3 & - 2 - 4 & 4 & - 3 \end{matrix} ⎤ ⎥ ⎦ ⎞ ⎟ ⎠$
$\Rightarrow Q = \frac{1}{2} ⎡ ⎢ ⎣ \begin{matrix} 0 & - 1 & - 5 1 & 0 & 6 5 & - 6 & 0 \end{matrix} ⎤ ⎥ ⎦$
$\Rightarrow Q = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 0 & \frac{- 1}{2} & \frac{- 5}{2} \frac{1}{2} & 0 & 3 \frac{5}{2} & - 3 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$
$Q^{'} = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 0 & \frac{1}{2} & \frac{5}{2} \frac{- 1}{2} & 0 & - 3 \frac{- 5}{2} & 3 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = - ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 0 & \frac{- 1}{2} & \frac{- 5}{2} \frac{1}{2} & 0 & 3 \frac{5}{2} & - 3 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = - Q$
$∴ Q^{'} = - Q$
$∴ Q$ is a skew symmetric matrix.
Now,
$P + Q = \frac{1}{2} (B + B^{'}) + \frac{1}{2} (B - B^{'}) = B$ Thus, $B$ is a sum of symmetric and a skew symmetric matrix.
Hence,
$\frac{⎡ ⎢ ⎣ \begin{matrix} 2 & - 2 & - 4 - 1 & 3 & 4 1 & - 2 & - 3 \end{matrix} ⎤ ⎥ ⎦}{B Given Matrix}$ $\frac{= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 2 & \frac{- 3}{2} & \frac{- 3}{2} \frac{- 3}{2} & 3 & 1 \frac{- 3}{2} & 1 & - 3 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦}{P Symmetric Matrix}$ $\frac{+ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 0 & \frac{- 1}{2} & \frac{- 5}{2} \frac{1}{2} & 0 & 3 \frac{5}{2} & - 3 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦}{Q Skew Symmetric Matrix}$

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