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.

Efan Halliday 2021-01-02 Answered
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.
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SchulzD
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,
$A=\left[\begin{array}{cc}1& -1\\ 0& 0\end{array}\right]$
$B=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$
Consider that there is an invertible matrix,
$P=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$ Step 2
The matrix A is similar to B as shown below,
$\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]={\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]}^{-1}\left[\begin{array}{cc}1& -1\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$
$=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$
$=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$
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,
$\sum _{i,j=1}^{n}|{b}_{ij}{|}^{2}\ne \sum _{i,j=1}^{n}|{a}_{ij}{|}^{2}$
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.
Jeffrey Jordon