Linear Algebra, Theory And Applications, 2012a

336 SELF ADJOINT OPERATORS
Now argue U is unitary and use this to establish the result. To show this verify
each row has length 1 and the inner product of two different rows gives 0. Now
U kj = e −i 2π n jk and so (U ∗ ) kj
= e i 2π n jk .
18. Let f be a periodic function having period 2π. The Fourier series of f is an expression
of the form
∞∑
n∑
c k e ikx ≡ lim c k e ikx
k=−∞
n→∞
k=−n
and the idea is to find c k such that the above sequence converges in some way to f. If
f (x) =
∞∑
k=−∞
c k e ikx
and you formally multiply both sides by e −imx and then integrate from 0 to 2π,
interchanging the integral with the sum without any concern for whether this makes
sense, show it is reasonable from this to expect
c m = 1
2π
∫ 2π
0
f (x) e −imx dx.
Now suppose you only know f (x) at equally spaced points 2πj/n for j =0, 1, ··· ,n.
Consider the Riemann sum for this integral obtained from using the left endpoint of
the subintervals determined from the partition { 2π
n j} n
. How does this compare with
j=0
the discrete Fourier transform? What happens as n →∞to this approximation?
19. Suppose A is a real 3 × 3 orthogonal matrix (Recall this means AA T = A T A = I. )
having determinant 1. Show it must have an eigenvalue equal to 1. Note this shows
there exists a vector x ≠ 0 such that Ax = x. Hint: Show first or recall that any
orthogonal matrix must preserve lengths. That is, |Ax| = |x| .
20. Let A be a complex m × n matrix. Using the description of the Moore Penrose inverse
in terms of the singular value decomposition, show that
lim
δ→0+ (A∗ A + δI) −1 A ∗ = A +
where the convergence happens in the Frobenius norm. Also verify, using the singular
value decomposition, that the inverse exists in the above formula.
21. Show that A + =(A ∗ A) + A ∗ . Hint: You might use the description of A + in terms of
the singular value decomposition.