Question

TRUE OR FALSE

Determine whether the statement “The matrix $\mathit{A}$ is similar to B is orthogonal, then B must be orthogonal as well.

Short answer

The given statement is false.

Step-by-step solution

Definition of the orthogonal matrix

The matrix $\mathit{A}$is orthogonal if and only if ${\mathit{A}}^{\mathbf{T}}\mathbf{=}{\mathit{A}}^{\mathbf{-}\mathbf{1}}$.

Explanation of the solution

Consider the statement as follows:

“If the matrix A is similar to B is orthogonal, then B must be orthogonal as well”.

Let P be any invertible matrix.

Since matrix A is similar to B and P is invertible

$A=P^{-1}BP$ ……(1)

Since A is orthogonal as follows.

$$\begin{gathered} A^T=A^{-1} \\ A{A}^T=A{A}^{-1} \\ A{A}^T=I \end{gathered}$$

Using equation (1) in the above equation as follows.

$$\begin{gathered} P^{-1}BP{\left(P^{-1}BP\right)}^T=I \\ {P}^{-1}BP{P}^T{B}^T{\left(P^{-1}\right)}^T=I \left(2\right) \end{gathered}$$

If P is orthogonal, then $P^T=P^{\_1},P{P}^T=I$and ${\left(P^{-1}\right)}^T=P$

Using the condition in equation (2) as follows.

$P^{T}B{B}^{T}P=1$

This implies that B is an orthogonal matrix.

Thus, B is the orthogonal matrix only if P is the orthogonal matrix.

Hence, the given statement “If the matrix A is similar to B is orthogonal, then B must be orthogonal as well” is false.