Math 4233
Homework Set 1
1. Let x=x1
x2and y=y1
y2be two complex vectors. Show that
(x,y) = (y,x)
2. If Ais a hermitian 2 ×2 matrix and xand yare 2-dimensional complex vectors as above, show that
(Ax,y)=(x,Ay)
2. Find the inverses of the following matrices:
(a) 1 4
−2 3
(b)
123
245
356
(c) 1i
i1
3. Find the eigenvectors and eigenvalues of the following matrices:
(a) 5−1
3 1
(b) 1i
i1
(c)
1 0 0
2 1 −2
3 2 1
4. For each of the matrices Ain Problem (3) find a diagonal matrix Dand an invertible matrix Csuch
that
D=C−1AC
1