Thread: diagonalizable

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?

Originally Posted by alexandrabel90

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



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...)

shouldnt
(1 2
-2i 1) =B?

Originally Posted by alexandrabel90

shouldnt
(1 2
-2i 1) =B?

Yes, sorry, the minus sign went walkabout between the LaTeX and the Ascii stuff. What did you get when you calculated AB and BA?

i got that

(5 2+2i
2-2i 5) = AB

(5 2i+2
2i+2 5)=BA

thus dont commute but it should commute

Originally Posted by alexandrabel90

i got that

(5 2+2i
2-2i 5) = AB

(5 2i+2
2i+2 5)=BA

thus dont commute but it should commute

You BA is incorrect, but your AB is correct. Looks like a simple case of a wandering minus sign to me...