MATH 310
Homework due 07/27/2011
1. Show the polarization identity for real inner product spaces:
<u,v>=1
4||u+v||2− ||u−v||2
2. Show the polarization identity for complex inner product spaces:
<u,v>=1
4||u+v||2− ||u−v||2+i||u+iv||2−i||u−iv||2
(a very neat way to rewrite this identity is:
<u,v>=1
4 3
X
k=0
ik||u+ikv||2!
)
3. Show that if Ais a Hermitian matrix then < Av,v>is a real number.
4. Diagonalize the matrix:
110
111
011
5. A Hermitian matrix is called positive if < Av,v>≥0 for every column
vector v. Show that A Hermitian matrix is positive if and only if its
eigenvalues are nonnegative real numbers.
1