Name: Understanding Trace, Transpose, and Conjugate of Matrix
Uploaded: 2019-07-19
Duration: 20 min 56 s
Description: Trace, transpose, and conjugate are the three important terms often used with respect to matrices. In this video, we will go through these terminologies and...

Question

Express the following matrix as the sum of a symmetric and skew-symmetric matrix and verify your result:

[\begin{matrix} 3 & - 2 & - 4 \\ 3 & - 2 & - 5 \\ - 1 & 1 & 2 \end{matrix}]

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Accepted Answer

Here, $A = [\begin{matrix} 3 & - 2 & - 4 \\ 3 & - 2 & - 5 \\ - 1 & 1 & 2 \end{matrix}]$

$\Rightarrow A^{T} = [\begin{matrix} 3 & 3 & - 1 \\ - 2 & - 2 & 1 \\ - 4 & - 5 & 2 \end{matrix}]$

Every square matrix $A$ can uniquely be expressed as the sum of a symmetric and skew-symmetric matrix i.e., $A = [\frac{1}{2} (A + A^{T})] + [\frac{1}{2} (A - A^{T})]$

Let $X = \frac{1}{2} (A + A^{T}) = \frac{1}{2} ([\begin{matrix} 3 & - 2 & - 4 \\ 3 & - 2 & - 5 \\ - 1 & 1 & 2 \end{matrix}] + [\begin{matrix} 3 & 3 & - 1 \\ - 2 & - 2 & 1 \\ - 4 & - 5 & 2 \end{matrix}])$

$\Rightarrow X = \frac{1}{2} [\begin{matrix} 6 & 1 & - 5 \\ 1 & - 4 & - 4 \\ - 5 & - 4 & 4 \end{matrix}]$

$\Rightarrow X = [\begin{matrix} 3 & \frac{1}{2} & \frac{- 5}{2} \\ \frac{1}{2} & - 2 & - 2 \\ \frac{- 5}{2} & - 2 & 2 \end{matrix}]$

Now, $X^{T} = {[\begin{matrix} 3 & \frac{1}{2} & \frac{- 5}{2} \\ \frac{1}{2} & - 2 & - 2 \\ \frac{- 5}{2} & - 2 & 2 \end{matrix}]}^{T} = [\begin{matrix} 3 & \frac{1}{2} & \frac{- 5}{2} \\ \frac{1}{2} & - 2 & - 2 \\ \frac{- 5}{2} & - 2 & 2 \end{matrix}] = X$
Let $Y = \frac{1}{2} (A - A^{T}) = \frac{1}{2} ([\begin{matrix} 3 & - 2 & - 4 \\ 3 & - 2 & - 5 \\ - 1 & 1 & 2 \end{matrix}] - [\begin{matrix} 3 & 3 & - 1 \\ - 2 & - 2 & 1 \\ - 4 & - 5 & 2 \end{matrix}]) = [\begin{matrix} 0 & \frac{- 5}{2} & \frac{- 3}{2} \\ \frac{5}{2} & 0 & - 3 \\ \frac{3}{2} & 3 & 0 \end{matrix}]$
$Y^{T} = {[\begin{matrix} 0 & \frac{- 5}{2} & \frac{- 3}{2} \\ \frac{5}{2} & 0 & - 3 \\ \frac{3}{2} & 3 & 0 \end{matrix}]}^{T} = [\begin{matrix} 0 & \frac{5}{2} & \frac{3}{2} \\ \frac{- 5}{2} & 0 & 3 \\ \frac{- 3}{2} & - 3 & 0 \end{matrix}] = - [\begin{matrix} 0 & \frac{- 5}{2} & \frac{- 3}{2} \\ \frac{5}{2} & 0 & - 3 \\ \frac{3}{2} & 3 & 0 \end{matrix}]$

Thus, $X$ is a symmetric matrix and $Y$ is a skew - symmetric matrix .
Now,
$X + Y = [\begin{matrix} 3 & \frac{1}{2} & \frac{- 5}{2} \\ \frac{1}{2} & - 2 & - 2 \\ \frac{- 5}{2} & - 2 & 2 \end{matrix}] + [\begin{matrix} 0 & \frac{- 5}{2} & \frac{- 3}{2} \\ \frac{5}{2} & 0 & - 3 \\ \frac{3}{2} & 3 & 0 \end{matrix}] = [\begin{matrix} 3 & - 2 & - 4 \\ 3 & - 2 & - 5 \\ - 1 & 1 & 2 \end{matrix}] = A$

Express the following matrix as the sum of a symmetric and skew-symmetric matrix and verify your result: $[\begin{matrix} 3 & - 2 & - 4 \\ 3 & - 2 & - 5 \\ - 1 & 1 & 2 \end{matrix}]$ .

Important Questions on Algebra of Matrices

Learn and Prepare for any exam you want !

Express the following matrix as the sum of a symmetric and skew-symmetric matrix and verify your result: 3-2-43-2-5-112.

Important Questions on Algebra of Matrices

Learn and Prepare for any exam you want !

Express the following matrix as the sum of a symmetric and skew-symmetric matrix and verify your result: $[\begin{matrix} 3 & - 2 & - 4 \\ 3 & - 2 & - 5 \\ - 1 & 1 & 2 \end{matrix}]$ .