Question

Use the various characterizations of orthogonal transformations and orthogonal matrices. Find the matrix of an orthogonal projection. Use the properties of the transpose. Which of the matrices in Exercise 1 through 4 are orthogonal? $\left[\begin{array}{cc}0.6& 0.8\\ 0.8& 0.6\end{array}\right]$.

Short answer

The Matrix $\left[\begin{array}{cc}0.6& 0.8\\ 0.8& 0.6\end{array}\right]$is not orthogonal.

Step-by-step solution

Definition of Orthogonal.

A square matrix is orthogonal matrix if $\mathbf{A}\mathbf{\times }{\mathbf{A}}^{\mathbf{T}}\mathbf{=}\mathbf{I}\mathbf{.}$

Verification whether the given matrix is orthogonal.

Let the given matrix is$A=\left[\begin{array}{cc}0.6& 0.8\\ 0.8& 0.6\end{array}\right]$ .

Then, the transpose of the given matrix is$A^T=\left[\begin{array}{cc}0.6& 0.8\\ 0.8& 0.6\end{array}\right]$.

Thus,for orthogonal matrix,

$$\begin{gathered} A\times A^T=\left[\begin{array}{cc}0.6& 0.8\\ 0.8& 0.6\end{array}\right]\times \left[\begin{array}{cc}0.6& 0.8\\ 0.8& 0.6\end{array}\right] \\ =\left[\begin{array}{cc}0.6& \times 0.6+\\ 0.8& \times 0.6+\end{array}\begin{array}{cc}0.8& 0.8\\ 0.6& 0.8\end{array}\begin{array}{cc}0.6& 0.8+\\ 0.8& 0.8+\end{array}\begin{array}{cc}0.8& 0.6\\ 0.6& 0.6\end{array}\right] \\ =\left[\begin{array}{cc}1& 0.96\\ 0.96& 1\end{array}\right] \end{gathered}$$

Since, the matrix not satisfy orthogonal conditionit gives$A\times A^T\ne I.$ .

Hence,$A=\left[\begin{array}{cc}0.6& 0.8\\ 0.8& 0.6\end{array}\right]$ is not an orthogonal matrix.