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?

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- Apr 15th 2011, 07:06 AMalexandrabel90diagonalizable
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? - Apr 15th 2011, 07:27 AMSwlabr
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...) - Apr 15th 2011, 07:31 AMalexandrabel90
shouldnt

(1 2

-2i 1) =B? - Apr 15th 2011, 07:34 AMSwlabr
- Apr 15th 2011, 07:50 AMalexandrabel90
i got that

(5 2+2i

2-2i 5) = AB

(5 2i+2

2i+2 5)=BA

thus dont commute but it should commute :( - Apr 15th 2011, 07:53 AMSwlabr