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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- April 15th 2011, 08:06 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? - April 15th 2011, 08:27 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

and

(1 2)

(-2i 1)=B

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...) - April 15th 2011, 08:31 AMalexandrabel90
shouldnt

(1 2

-2i 1) =B? - April 15th 2011, 08:34 AMSwlabr
- April 15th 2011, 08:50 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 :( - April 15th 2011, 08:53 AMSwlabr