Homework 1. (Due Jan. 21)
Math. 639
In this assignment, K is R or C and
(x, y) =
n
X
xi ȳi , for x, y ∈ K n .
i=1
Problem 1. Using the complex version of the spectral theorem, prove its
corollary. (Hint: First prove that real and imaginary parts of a complex
eigenvector are eigenvectors (with the original eigenvalue) when the matrix
is real valued.)
Problem 2. An n × n matrix A with values in K is positive definite if
(Ax, x) is real valued and non-negative for every x ∈ K n and only zero for
x = 0.
(a) Show that the eigenvalues of a positive definite matrix are real and positive.
(b) Show that if < ·, · > is an inner product on K n then
(2.1)
< x, y >= (M x, y) for all x, y ∈ K n
where M is a symmetric (resp. Hermitian) positive definite matrix when
K = R (resp. K = C). (The converse is also true but you need not prove
it.)
Problem 3. An n × n matrix A on K is “self adjoint with respect to an
inner product < ·, · >” if
< Ax, y >=< x, Ay > for all x, y ∈ K n .
Show that such a matrix A has real eigenvalues and there is a < ·, · >orthonormal basis of eigenvectors for A when K = C. (Hint: Let M be
e = M 1/2 AM −1/2 is Hermitian by showing that
as in (2.1). Show that A
e y) = (x, Ay)
e for all x, y ∈ K n . Apply the original spectral theorem to A.
e
(Ax,
Obtain the < ·, · > orthonormal eigenvectors of A from the (·, ·)-orthonormal
e Note that the square root of M is defined in terms of its
eigenvectors of A.
orthonormal (with respect to (·, ·)) basis of eigenvectors, {ψ1 , . . . , ψn }, i.e. if
x=
n
X
i=1
1
c i ψi
2
then
M
1/2
x=
n
X
i=1
1/2
c i λi ψ i
and M
−1/2
x=
n
X
−1/2
c i λi
ψi .
i=1
Here λi is the eigenvalue corresponding to ψi . It is easy to see that M 1/2 M −1/2 =
I and M 1/2 M 1/2 = M and that both M 1/2 and M −1/2 are Hermitian and
positive definite. )
Remark 1. Using the same technique as in Problem 1, the above result can
be extended to real “self adjoint” matrices, i.e., the orthonormal eigenvectors
can be taken to be in Rn .