Math 477 — Homework Assignment 5, due Nov.9, 2006
1. For each of the following statements, prove that it is true or give an example to show it is false.
Throughout, A∈Cm×munless otherwise noted.
(a) If λis an eigenvalue of Aand µ∈C, then λ−µis an eigenvalue of A−µI.
(b) If Ais real and λis an eigenvalue of A, then so is −λ.
(c) If Ais real and λis an eigenvalue of A, then so is λ.
(d) If λis an eigenvalue of Aand Ais nonsingular, then λ−1is an eigenvalue of A−1.
2. Find the Schur factorizations of
A=3 8
−2 3 and B=4 7
1 12 .
3. Here is Gerschgorin’s theorem, which holds for any m×mmatrix A, symmetric or nonsymmetric:
Every eigenvalue of Alies in at least one of the mcircular disks in the complex plane
with centers aii and radii Pj6=i|aij |. Moreover, if nof these disks form a connected
domain that is disjoint from the other m−ndisks, then there are precisely neigenvalues
of Awithin this domain.
(a) Prove the first part of Gerschgorin’s theorem. (Hint: Let λbe any eigenvalue of A, and x
a corresponding eigenvector normalized so that its largest entry is 1.)
(b) Give estimates based on Gerschgorin’s theorem for the eigenvalues of
A=
810
1 4 ε
0ε1
,|ε|<1.
4. Suppose we have a 3×3 matrix and wish to introduce zeros by left- and/or right-multiplications
by unitary matrices Qjsuch as Householder reflections or Givens rotations. Consider the fol-
lowing three matrix structures:
(a)
x x 0
0x x
0 0 x
,(b)
x x 0
x0x
0x x
,(c)
x x 0
0 0 x
0 0 x
.
For each one, decide which of the following situations holds, and justify your claim.
(i) Can be obtained by a sequence of left-multiplications by matrices Qj;
(ii) Not (i), but can be obtained by a sequence of left- and right-multiplications by matrices
Qj;
(iii) Cannot be obtained by any sequence of left- and right-multiplications by matrices Qj.
5. Let A∈Cm×mbe given, not necessarily Hermitian. Show that a number z∈Cis a Rayleigh
quotient of Aif and only if it is a diagonal entry of Q∗AQ for some unitary matrix Q. Thus
Rayleigh quotients are just diagonal entries of matrices, once you transform orthogonally to the
right coordinate system.