Symmetric, Skew-Symmetric And Orthogonal Matrices

Suppose I have a matrix $\textbf{A}$ . Then the matrix will be symmetric if the transpose of the matrix $\textbf{A}$ is the same as the original matrix.

Now I give an example.

Let’s say I have a matrix $\textbf{A}$ like

$\textbf{A} = \begin{pmatrix} a_{11}&a_{12}\\ a_{21}&a_{22} \end{pmatrix}.$

Now I’ll check if it’s a symmetric matrix.

First of all, I’ll transpose the matrix. And this means the first row will be the first column. Then the second row will be the second column. And it will continue in this way.

So the transpose of the matrix $\textbf{A}$ is $\textbf{A}^\textbf{T}$ . Thus the value of $\textbf{A}^\textbf{T}$ will be

$\textbf{A}^\textbf{T} = \begin{pmatrix} a_{11}&a_{21}\\ a_{12}&a_{22} \end{pmatrix}.$

Now the matrix $\textbf{A}$ will be

$\boxed{\text{symmetric if}~\textbf{A} = \textbf{A}^\textbf{T},}$

$\boxed{\text{skew-symmetric if}~\textbf{A} = -\textbf{A}^\textbf{T}}$

and

$\boxed{\text{orthogonal if}~\textbf{A}^\textbf{T} = \textbf{A}^{-1}.}$

Okay, now I’ll give five different examples on symmetric, skew-symmetric and orthogonal matrices.

EXAMPLE 1

According to Kreyszig (2005)*, “Is the following matrix symmetric, skew-symmetric, or orthogonal?

$\begin{pmatrix} 3&1\\ -1&1 \end{pmatrix}.$

”

SOLUTION

Now here the matrix is

$\begin{pmatrix} 3&1\\ -1&1 \end{pmatrix}.$

First of all, I’ll give it a name, say $\textbf{A}$ . So it will be

$\textbf{A} = \begin{pmatrix} 3&1\\ -1&1 \end{pmatrix}.$

Now I’ll check whether it’s a symmetric, skew-symmetric or orthogonal matrix.

STEP 1

First of all, I’ll transpose the matrix $\textbf{A}$ . So the first row becomes the first column. Then the second row will be the second column. In the end, the third row will be the third column.

Thus the matrix $\textbf{A}^\textbf{T}$ is

$\textbf{A}^\textbf{T} = \begin{pmatrix} 3&-1\\ 1&1 \end{pmatrix}.$

As I can see, this is neither $\textbf{A}$ nor $-\textbf{A}$ , that is

$\boxed{\textbf{A}^\textbf{T} \neq \textbf{A}~\text{or}~\textbf{A}^\textbf{T} \neq-\textbf{A}.}$

So that means the matrix $\textbf{A}$ is neither a symmetric or a skew-symmetric matrix.

Next, I’ll check if it’s an orthogonal matrix. Now for that, I’ll get the inverse of the matrix $\textbf{A}$ .

STEP 2

But I won’t use the standard way to get the inverse of the matrix. Instead, I’ll use the Gauss-Jordan method to find out the inverse of the matrix $\textbf{A}$ .

As per the Gauss-Jordan method, I’ll include the unit matrix $\textbf{I}$ on the right-hand side like

$[\textbf{A}\qquad \textbf{I}]= \begin{pmatrix} ~3 & 1&|&1&0\\ -1&1&|&0&1 \end{pmatrix}.$

And my aim is to bring the unit matrix on the left-hand side. Also the matrix on the right-hand side will be the inverse of the matrix $\textbf{A}$ .

Ok, I’ll start now.

First of all, I’ll interchange row 1 and row 2 to get the equivalent matrix as

$[\textbf{A}\qquad \textbf{I}] = \begin{pmatrix} -1 & 1&|&0&1\\ ~3&1&|&1&0 \end{pmatrix}.$

Now I’ll add three times row 1 with row 2. And in mathematical terms, it will be

Row 1 + 3 (Row 2).

So this gives the equivalent matrix as

$[\textbf{A}\qquad \textbf{I}] = \begin{pmatrix} -1 & 1&|&0&1\\ ~0&4&|&1&3\end{pmatrix}.$

Next, I’ll divide row 2 by four. Simultaneously, I’ll also multiply row 1 with (-1) . And in mathematical terms, it will be

(-1) Row 1, (1/4) Row 2.

Hence the equivalent matrix matrix will be

$[\textbf{A}\qquad \textbf{I}] = \begin{pmatrix} 1 & -1&|&0&-1\\ 0&~1&|&1/4&3/4\end{pmatrix}.$

Then I’ll add row 2 to row 1. And in mathematical terms, it will be

Row 1 + Row 2.

So the equivalent matrix is

$[\textbf{A}\qquad \textbf{I}] = \begin{pmatrix} 1 & 0&|&1/4&-1/4\\ 0&1&|&1/4&~3/4 \end{pmatrix}.$

Now I have already got the unit matrix on the left-hand side. So that means the inverse of the matrix $\textbf{A}$ is

$\textbf{A}^{-1} = \begin{pmatrix} 1/4&-1/4\\ 1/4&~3/4\end{pmatrix}.$

As I can see, this is not the matrix $\textbf{A}^\textbf{T}$ , that is

$\boxed{\textbf{A}^\textbf{T} \neq \textbf{A}^{-1}.}$

So the conclusion is that the given matrix is not an orthogonal matrix either.

CONCLUSION

As I have seen from Step 1, the given matrix is neither a symmetric or a skew-symmetric one. Also, from Step 2, I have seen that the given matrix is not an orthogonal one either.

So the conclusion is that this specific matrix is none of the symmetric, skew-symmetric or orthogonal matrices. And this is the answer to the given example.