Diagonalizing a Complex Matrix: Unitary Matrix Approach

[tas3113]

I am going over old practice exams and came across this question:

Find the unitary matrix which diagonalizes the matrix

...i 1 0
A = (-1 i 0 )
... 0 0 -i


First off, can someone explain to me about unitary matrices and get me started on this question? I do not know where to begin solving it. Thank you!

 

[Cyosis]

A unitary matrix is a matrix for which U^{\dagger}U^{-1}=U^{-1}U^\dagger=I. So U^\dagger=U^{-1}. A matrix B is diagonalizable if there exists a matrix C such that C^{-1}BC is diagonal.

 

[tas3113]

so i found the conjugate transpose of the matrix and got

(-i -1 0)
(1 -i 0)
(0 0 i)

is that a start?

 

[Cyosis]

No, the idea is that you find U^\dagger A U. A need not be unitary. The first step is to find the eigenvalues of A and their respective eigenvectors. You can however use A^\dagger to check if A is unitarily diagonalizable. A matrix A is unitarily diagonalizable if and only if A^\dagger A=A A^\dagger. Perhaps it's nice to check this for A, although of course A is unitarily diagonalizable.

After you have the eigenvectors check if they form an orthonormal set. If they don't you have to make an orthonormal set by the Gram-Schmidt process. Now make a matrix with its columns being the eigenvectors.

 

[tas3113]

so i came up with the equation:
-h(h^2+1)+h^2 i

so the eigenvalues are 0, -i, and something else but i don't know how to find that. 0 = ih^2

the eigenvectors are (1,- i,0) , (0,0,1), and something..

 

[Cyosis]

That equation doesn't look right to me. I got (-i-\lambda)[(i-\lambda)(i-\lambda)+1]=0

 

[tas3113]

so its
<br /> \left|\begin{array}{ccc}i-\lambda &amp; 1 &amp; 0 \\ -1 &amp; i-\lambda &amp; 0 \\ 0 &amp; 0 &amp; -i-\lambda\end{array}\right|<br />

so then
<br /> [(i-\lambda)(i-\lambda)(-i-\lambda)]-[(-1)(1)(-i-\lambda)]=0<br />
oo.. i forgot the lambda for the last (-i) before.. ok so
<br /> (i-\lambda)(i-\lambda)(-i-\lambda)-(i+\lambda)=0<br />
and so
<br /> (-i-\lambda)[(i-\lambda)(i-\lambda)+1]=0<br />
is right

so eigenvalues are 0, -i, 2i
with eigenvectors: (1,- i,0), (0,0,1), (1, i,0)

ok... so as for orthonormal set:
(1,- i,0), (0,0,1) are perpendicular
(0,0,1), (1, i,0) are perpendicular

then? lost from there

 

[Cyosis]

Well this set of eigenvectors is orthogonal and we want an orthonormal set. So you need to normalize them. After that you can simply form a matrix with the eigenvectors as columns. Then test if it's unitary and diagonalize the matrix A.

A tip when it comes to calculating determinants. You don't always have to start from the top left and work to the top right when you write a 3x3 determinant as the sum of 2x2 determinants. Like in this case the fastest way is:

<br /> \left|\begin{array}{ccc}i-\lambda &amp; 1 &amp; 0 \\ -1 &amp; i-\lambda &amp; 0 \\ 0 &amp; 0 &amp; -i-\lambda\end{array}\right|=(-i-\lambda)\left|\begin{array}{cc}i-\lambda &amp; 1 \\ -1 &amp; i-\lambda &amp; \end{array}\right|<br />

Start at the bottom right, because that row/column has the most zeros.

 

[tas3113]

aren't they already normalized?
u1 = (1,- i,0)
u2 = (0,0,1) - (0+0+0)/... doesn't matter
u3 = (1, i,0) - (0+0+0)/... doesn't matter

so then i would divide by ||u||
u1: squareroot(1^2+(-i)^2+0) = 0 --well then itd be undef if u divide by 0
u2: 1
u3: 0

 

[Cyosis]

A normalized vector has length 1. So ||u_1||=\sqrt{(u_1,u_1)}=\sqrt{2}. Normalizing u1 gives 1/\sqrt{2}(1,-i,0). It works the same for the others.

 

[tas3113]

ok. i ended up with the vectors
<br /> (1/\sqrt{2}, -i/\sqrt{2}, 0), (0, 0, 1), (1/\sqrt{2}, i/\sqrt{2}, 0)<br />

which gives me the matrix
<br /> \left|\begin{array}{ccc}1/\sqrt{2} &amp; 0 &amp; 1/\sqrt{2} \\ -i/\sqrt{2} &amp; 0 &amp; i/\sqrt{2} \\ 0 &amp; 1 &amp; 0\end{array}\right|<br />

and the conjugate transpose is
<br /> \left|\begin{array}{ccc}1/\sqrt{2} &amp; i/\sqrt{2} &amp; 0 \\ 0 &amp; 0 &amp; 1 \\ 1/\sqrt{2} &amp; -i/\sqrt{2} &amp; 0\end{array}\right|<br />

then
<br /> \left|\begin{array}{ccc}1/\sqrt{2} &amp; 0 &amp; 1/\sqrt{2} \\ -i/\sqrt{2} &amp; 0 &amp; i/\sqrt{2} \\ 0 &amp; 1 &amp; 0\end{array}\right|<br /> * <br /> \left|\begin{array}{ccc}1/\sqrt{2} &amp; i/\sqrt{2} &amp; 0 \\ 0 &amp; 0 &amp; 1 \\ 1/\sqrt{2} &amp; -i/\sqrt{2} &amp; 0\end{array}\right|<br /> = <br /> \left|\begin{array}{ccc}1 &amp; 0 &amp; 0 \\ 0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right|<br />
which is already diagonalized

so then the unitary matrix/answer is:
<br /> <br /> \left|\begin{array}{ccc}1/\sqrt{2} &amp; i/\sqrt{2} &amp; 0 \\ 0 &amp; 0 &amp; 1 \\ 1/\sqrt{2} &amp; -i/\sqrt{2} &amp; 0\end{array}\right|<br /> <br />
?

 

[Cyosis]

Yep that is correct. You've shown that U^\dagger U=I. Therefore U is unitary. You can test if U indeed diagonalizes A by calculating U^\dagger A U.

 

[tas3113]

ok. thank you!