Question

Suppose that A and B are diagonalizable matrices. Prove or disprove that A is similar to B if and only if A and B are unitarily equivalent.

Answer 1

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.