Step 1

Consider A and B are matrices that are diagonalizable.

The objective is to prove or disprove that A is similar to B only if A and B are unitarily equivalent.

Consider the two diagonalizable matrices are,

\(A=\begin{bmatrix}1& -1 \\ 0 & 0 \end{bmatrix}\)

\(B=\begin{bmatrix}1& 0 \\ 0 & 0 \end{bmatrix}\)

Consider that there is an invertible matrix,

\(P=\begin{bmatrix}1& 1 \\ 0 & 1 \end{bmatrix}\) Step 2

The matrix A is similar to B as shown below,

\(\begin{bmatrix}1& 0 \\ 0 & 0 \end{bmatrix}=\begin{bmatrix}1& 1 \\ 0 & 1 \end{bmatrix}^{-1} \begin{bmatrix}1& -1 \\ 0 & 1 \end{bmatrix}\begin{bmatrix}1& 1 \\ 0 & 1 \end{bmatrix}\)

\(=\begin{bmatrix}1& 1 \\ 0 & 1 \end{bmatrix}\begin{bmatrix}1& 0 \\ 0 & 0 \end{bmatrix}\)

\(=\begin{bmatrix}1& 0 \\ 0 & 0 \end{bmatrix}\)

The A and B are similar matrices.

But A and B are not unitary because B is symmetric, but A is not.

Also, recall the result, If B and A are unitarily equivalent.

\(\sum_{i,j=1}^{n}|b_{ij}|^2=\sum_{i,j=1}^{n}|a_{ij}|^2\)

Since, 2 is not equal then,

\(\sum_{i,j=1}^{n}|b_{ij}|^2\neq\sum_{i,j=1}^{n}|a_{ij}|^2\)

Since,

A and B are not unitarily equivalent.

Therefore, it is not necessary that A is similar to B if and only if A and B are unitarily equivalent.

Consider A and B are matrices that are diagonalizable.

The objective is to prove or disprove that A is similar to B only if A and B are unitarily equivalent.

Consider the two diagonalizable matrices are,

\(A=\begin{bmatrix}1& -1 \\ 0 & 0 \end{bmatrix}\)

\(B=\begin{bmatrix}1& 0 \\ 0 & 0 \end{bmatrix}\)

Consider that there is an invertible matrix,

\(P=\begin{bmatrix}1& 1 \\ 0 & 1 \end{bmatrix}\) Step 2

The matrix A is similar to B as shown below,

\(\begin{bmatrix}1& 0 \\ 0 & 0 \end{bmatrix}=\begin{bmatrix}1& 1 \\ 0 & 1 \end{bmatrix}^{-1} \begin{bmatrix}1& -1 \\ 0 & 1 \end{bmatrix}\begin{bmatrix}1& 1 \\ 0 & 1 \end{bmatrix}\)

\(=\begin{bmatrix}1& 1 \\ 0 & 1 \end{bmatrix}\begin{bmatrix}1& 0 \\ 0 & 0 \end{bmatrix}\)

\(=\begin{bmatrix}1& 0 \\ 0 & 0 \end{bmatrix}\)

The A and B are similar matrices.

But A and B are not unitary because B is symmetric, but A is not.

Also, recall the result, If B and A are unitarily equivalent.

\(\sum_{i,j=1}^{n}|b_{ij}|^2=\sum_{i,j=1}^{n}|a_{ij}|^2\)

Since, 2 is not equal then,

\(\sum_{i,j=1}^{n}|b_{ij}|^2\neq\sum_{i,j=1}^{n}|a_{ij}|^2\)

Since,

A and B are not unitarily equivalent.

Therefore, it is not necessary that A is similar to B if and only if A and B are unitarily equivalent.