# If A is diagonalizable and for all eigenvalues , lambda text{ of } A , |lambda|=1 , then A is unitary. True or False?

If A is diagonalizable and for all eigenvalues , , then A is unitary. True or False?
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FieniChoonin
Step 1
Solution.
Given: A is diagonalizable and all eigenvalue
Step 2
Let $A=\left[\begin{array}{cc}1& 1\\ 0& -1\end{array}\right]$ , the eigenvalue of A is 1,-1 we know that if A has distinct eigenvalue then A is diagonalizable.
$\therefore$ A is diagonalizable , and 1,-1 is eigenvalue of A also $|1|=1,|-1|=1$
Now check A is unitary?
If A is unitary then $A{A}^{T}=I$ ,
$A{A}^{T}=\left[\begin{array}{cc}1& 1\\ 0& -1\end{array}\right]{\left[\begin{array}{cc}1& 1\\ 0& -1\end{array}\right]}^{T}=\left[\begin{array}{cc}1& 1\\ 0& -1\end{array}\right]\left[\begin{array}{cc}1& 0\\ 1& -1\end{array}\right]=\left[\begin{array}{cc}2& -1\\ -1& 1\end{array}\right]\ne \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$\therefore$ A is not unitary
Hence if A is diagonalizable and all eigenvalue

then A need not be unitary
The given statement is False
Jeffrey Jordon