MATH 223
Assignment #2
Due Friday September 25 at start of class
(two pages)
1.
a) Assume we have two nonzero vectors u, v. Let M = [u v], the 2 × 2 matrix with column 1
being u and column 2 being v. Show that the 2 × 2 matrix M = [u v] is invertible if and
only if u 6= pv for any real number p.
b) Assume A is a 2 × 2 matrix with two different eigenvalues λ1 6= λ2 and eigenvectors v1 , v2
(Av1 = λ1 v1 and Av2 = λ2 v2 ). Show that v1 6= pv2 for any real number p. Thus show that
the matrix M = [v1 v2 ] is invertible and thus show that A is diagonalizable.
2. Let Ak denote the 2 × 2 matrix
"
Ak =
k 1
−1 3
#
For what values of k is Ak diagonalizable? Namely, can we find two eigenvectors u, v of Ak so that
if we let M = [u v] then M is invertible? Use 1(b) to handle the case of two different eigenvalues;
you need not explicitly find the eigenvectors in that case.
"
#
"
#
5 −6
1 1
3. Let A1 =
and A2 =
. For each of these two matrices, determine the
1 0
1 3
eigenvalues and for each eigenvalue determine an eigenvector. For A2 the eigenvalues are a little
more complicated making the computations a little harder. Then give the diagonalization of each
matrix; namely an invertible matrix M and a diagonal matrix D with AM = M D (i.e. A =
M DM −1 and M −1 AM = D).
4. Recall that ex = 1 + x + 2!1 x2 + 3!1 x3 + 4!1 x4 + · · ·. Define
eA = I + A +
1 2 1 3 1 4
A + A + A + ···
2!
3!
4!
assuming that the infinite sum makes sense. We are generalizing exponentiation to matrices.
"
a) For D =
#
1.2 0
, compute eD .
0 2
"
#
2 6
b) For B = M DM where M =
, compute eB . (Use our matrix distributive law on an
1 2
infinite sum; we haven’t justified that this is reasonable but it is OK for you to use it here)
−1
5.
"
a) Let S =
#
λ 1
. It can be shown that S is not diagonalizable. Show (by induction) that
0 λ
"
St =
λ 1
0 λ
"
#
#t
"
=
λt tλt−1
0
λt
#
.
2 6
b) For C = M SM where M =
, compute C t . (The idea of the question is to show
1 2
that you don’t need the matrix to be diagonalizable in order to simplify powers.)
−1
6.
x ∈ R2 can be written uniquely as a linear combination of u1 , u2 where u1 =
Show that
3
−1
, u2 =
. Draw/sketch the grid associated with u1 , u2 coordinates on top of the usual x, y1
2
plane, say identifying the lines with u1 coordinates -1,0,1 and 2 and the lines with u2 coordinates
-1,0 and 1 as well as the lines with x coordinates -1,0,1 and the lines y coordinates -1,0,1. Different
colours for the u1 , u2 lines from the x, y lines would be helpful.
7. Let A be a 2 × 2 matrix with two different eigenvalues λ1 , λ2 and associated eigenvectors v1 , v2 .
Let v = av1 + bv2 . Assume that |λ1 | > |λ2 |. Show that
An v
= av1
n→∞ λn
1
lim
For a 6= 0 this means that we see the eigenevector v1 appearing in the limit.
8. Review the notes on Fibonacci numbers. Let f1 , f2 be two arbitrary integers, not both zero.
Consider the sequence f1 , f2 , f3 , f4 , . . . where fi = fi−1 + fi−2 for i = 3, 4, 5, . . .. We wish to show
that
√
1+ 5
fn
=
.
lim
n→∞ f
2
n−1
(My Linear Algebra instructor Harry F Davis in 1973, said that he was contacted by a member of
the public who had noticed this lovely fact) Firstly, explain why we can solve for c1 , c2 in the vector
equation
√ #
√ #
"
"
#
"
1− 5
1+ 5
f2
2
2
+ c2
.
= c1
f1
1
1
Using our hypothesis that f1 , f2 are not both zero, we deduce that c1 , c2 are not both
√ zero. Secondly,
use our hypothesis that f1 , f2 are integers to deduce c1 6= 0. The irrationality of 5 combined with
c1 , c2 being integers is important. Thirdly verify the limit. If you can’t show c1 6= 0 then go ahead
and assume c1 6= 0 to establish this limit.
Hint: use ideas of question 7.