Find a unitary matrix U so that for
A = [ 2, 3-3i] [3+3i, 5],A = UDU^H, where D is diagonal
Explain why the determinant of A has to have real determinant
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) =
$\displaystyle \begin{vmatrix}x-2&-3+3i\\-3-3i&5-x \end{vmatrix}$
$\displaystyle = (x - 2)(x - 5) - (-3+3i)(-3-3i) = x^2 - 7x + 10 - 18$
$\displaystyle = x^2 - 7x - 8 = (x - 8)(x + 1)$
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:
$\displaystyle U = (u,v)^* = \begin{bmatrix}\frac{-1-i}{\sqrt{3}}&\frac{1}{\sqrt{3}}\\ \frac{1+i}{\sqrt{6}}&\frac{2}{\sqrt{6}} \end{bmatrix}$
i leave it in your capable hands to show that UAU* =
$\displaystyle \begin{bmatrix}-1&0\\0&8 \end{bmatrix}$
which is, of course, diagonal.