Math410 (Bayly) Homework 5 (due Monday 29 March)
Solutions should be posted on Friday 26 March, so you can grade them over the weekend.
The ability to find eigenvalues and eigenvectors of matrices is of enormous importance in this course. I’m not going to ask you to do anything very lengthy, but you should be able to handle any 2x2 matrix I throw at you, plus the occasional 3x3 or 4x4 with some structure that makes the eigenproblem especially simple in some way.
I strongly suggest you try all these problems, not just the ones to be turned in, including the extra credit problems. They all illustrate interesting aspects of the theory which, even if not central, help the overall subject make more sense.
One aspect of eigenvalue theory
that Strang does not mention until the end of chapter
5 is the Cayley-Hamilton Theorem, which says tersely
“A matrix satisfies its own characteristic polynomial”. More precisely, if 
is any polynomial in a single scalar variable x , we can define
the same polynomial for a square matrix A
by 
. The CH theorem says
that if the polynomial is the char poly 
of the matrix A , then 
(the matrix of all zeros).
In every problem that involves finding the char poly of a matrix, I’d
like you to plug the matrix into the polynomial and check that the result is
indeed zero.
(WHY is the Cayley Hamilton
Theorem actually true? Think about
writing A in its diagonal form 
and then plugging into the polynomial. This is optional!)
Section 5.1: All problems; turn in (4), (7), and (14).
Section 5.2: All problems; turn in (2), (3), (6), (8), (13),
(14)
Section 5.3: Turn in problem (1).
EXTRA CREDIT
Eigenvalues and eigenvectors of projections, rotations, and reflections. We encountered these matrices in Chapter 3. Their special geometric properties are reflected in special eigenproperties.
Recall that a projection
is any matrix P with the property that 
.
(1) Show that det(P) must equal 0 or
1. Then show that any eigenvalue of P
must also be 0 or 1. What does it mean
for the eigenvector
if the eigenvalue is 0 or 1?
Rotations and reflections are
matrices Q with the property that 
. We showed in class
one day that 
(do so again, just for
the practice). If det(Q)=1 we call Q a rotation, while if det(Q)=-1 we call Q a reflection.
One of the properties of such
matrices is isometry,
i.e. the length of 
is the same as the
length of 
for any vector . We’re assuming the
matrix Q has all real components, but
when we’re considering eigenvectors and eigenvalues
we have to allow the possibility of complex numbers. Therefore the definition of length has to become 
, where the * indicates the complex conjugate of each
component of . Then if is an eigenvector of Q,
the corresponding eigenvalue
must satisfy
, i.e. the modulus (complex absolute value) of must be unity. A further restriction on the set of eigenvalues of comes from the fact that the characteristic
polynomial has all real coefficients if the matrix Q has all real entries.
Therefore any complex roots (eigenvalues) come
in conjugate pairs.
The foregoing remarks apply to all dimensions, but I’d like you to restrict attention to 3d space and 3x3 matrices for now.
(2) Show that for any 3x3 matrix A, 
, where 
are numbers determined
(but you don’t have to find them) by the elements of A. Argue that must have at least one
real root, which implies that every 3x3 matrix has at least one real eigenvalue.
(3) Show that every 3x3 rotation matrix has at least one eigenvalue equal to +1. What does this mean for the corresponding eigenvector?
(4) Show that every 3x3 reflection matrix has at least one eigenvalue equal to -1. What does this mean for the corresponding eigenvector?