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,
Consider that there is an invertible matrix,
The matrix A is similar to B as shown below,
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.
Since, 2 is not equal then,
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.