in my book, it states that the 2x2 matrix,
(1 2i
2 1) is unitarily diagonalizable. i thought it has to be normal to be unitarily diagonalizable? in this case, the matrix doesnt seem to be normal.
may i know what is wrong with my reasoning?
in my book, it states that the 2x2 matrix,
(1 2i
2 1) is unitarily diagonalizable. i thought it has to be normal to be unitarily diagonalizable? in this case, the matrix doesnt seem to be normal.
may i know what is wrong with my reasoning?
It is normal...(you should remember that a matrix is normal if and only if it is unitarily diagonalisable).
It is normal as if
(1 2i)
(2 1)=A
$\displaystyle A=\left( \begin{array}{cc}
1 & 2i \\
2 & 1 \end{array} \right)$
and
(1 2)
(-2i 1)=B
$\displaystyle B=\left( \begin{array}{cc}
1 & 2 \\
-2i & 1 \end{array} \right)$
then A and B commute. As B is the conjugate transpose thing of A, you are done...
(I've left the LaTeX in in the hope that the compiler will start working again...)