12.13. EXERCISES 323

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 makessense, show it is reasonable from this to expect

cm =1

2π

∫ 2π

0

f (x) e−imxdx.

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 ofthe subintervals determined from the partition

{2πn j}nj=0

. How does this compare with

the discrete Fourier transform? What happens as n→ ∞ to this approximation?

18. Suppose A is a real 3 × 3 orthogonal matrix (Recall this means AAT = ATA = I. )having determinant 1. Show it must have an eigenvalue equal to 1. Note this showsthere exists a vector x ̸= 0 such that Ax = x. Hint: Show first or recall that anyorthogonal matrix must preserve lengths. That is, |Ax| = |x| .

19. Let A be a complex m×n matrix. Using the description of the Moore Penrose inversein terms of the singular value decomposition, show that

limδ→0+

(A∗A+ δI)−1A∗ = A+

where the convergence happens in the Frobenius norm. Also verify, using the singularvalue decomposition, that the inverse exists in the above formula. Observe that thisshows that the Moore Penrose inverse is unique.

20. Show that A+ = (A∗A)+A∗. Hint: You might use the description of A+ in terms of

the singular value decomposition.

21. In Theorem 12.6.1. Show that every matrix which commutes with A also commuteswith A1/k the unique nonnegative self adjoint kth root.

22. Let X be a finite dimensional inner product space and let β = {u1, · · · , un} be anorthonormal basis for X. Let A ∈ L (X,X) be self adjoint and nonnegative andlet M be its matrix with respect to the given orthonormal basis. Show that M isnonnegative, self adjoint also. Use this to show that A has a unique nonnegative selfadjoint kth root.

23. Let A be a complex m × n matrix having singular value decomposition U∗AV =(σ 0

0 0

)as explained above, where σ is k × k. Show that

ker (A) = span (V ek+1, · · · , V en) ,

the last n− k columns of V .

24. The principal submatrices of an n × n matrix A are Ak where Ak consists thoseentries which are in the first k rows and first k columns of A. Suppose A is a realsymmetric matrix and that x →⟨Ax,x⟩ is positive definite. This means that if x ̸= 0,then ⟨Ax,x⟩ > 0. Show that each of the principal submatrices are positive definite.

Hint: Consider(

xT 0)A

(x

0

)where x consists of k entries.