note that A is hermetian, so A = A*.

note as well that A* = (UDU*)* = U**D*U* = UD*U*.

this means that D* = U*A*U = U*AU = D.

now det(A) = det(UDU*) = det(D). since D = diag(d_{11},d_{22}),

D* = diag(d_{11}*,d_{22}*), and det(D) = det(D*) thus means:

d_{11}d_{22}= (d_{11})*(d_{22})* = (d_{11}d_{22})*

hence det(A) = det(D) is real.

in this particular case, of course, one can just compute det(A):

det(A) = (2)(5) - (3+3i)(3-3i) = 10 + 9 - 9i^{2}= 10 + 9 - (-9) = 10 + 18 = 28

*****

the above proof can be adapted easily to any nxn Hermitian matrix.

so how do we find such a matrix U?

first, lets find the eigenvalues for A:

det(xI - A) =

so the eigenvalues of A are -1 and 8. solving (A + I)u = 0 we get:

u = (-1+i,1) as an eigenvector. normalizing u we get the unit vector ((-1+i)/√3,1/√3).

solving (A - 8I)v = 0, we get:

v = (1-i,2), which normalizes to the unit vector ((1-i)/√6,2/√6). note that <u,v> = 0, so we may take:

i leave it in your capable hands to show that UAU* =

which is, of course, diagonal.