Question

What is meant by orthogonal matrix?

Answer 1

orthogonal matrix:

• If the transpose of a square matrix with real entries is equal to its inverse, then the matrix is called an orthogonal matrix.
• $A$ is an orthogonal matrix, if $A^T=A^{-1}$.

Example: $A=\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]$

Then, the transpose of the matrix $A$ is,

$A^T=\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]$ $.....\left(1\right)$

The determinant of the matrix $A$ is,

$\left|A\right|=\left(-1\right)\left(1\right)-\left(0\right)\left(0\right)$

$\Rightarrow$ $\left|A\right|=-1$

And the adjoint matrix of $A$ is,

$adj.\left(A\right)=\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]$

Thus, the inverse matrix of $A$ is,

$A^{-1}=\frac{1}{1}\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]$

$\Rightarrow$ $A^{-1}=\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]$ $.....\left(2\right)$

From equations $\left(1\right)$ and $\left(2\right)$, we get

$A^T=A^{-1}$

Hence, the matrix $A$ is an orthogonal matrix,