6.3
4. The matrix is
0 -1
1 2
I will use x instead of lambda.
The characteristic polynomial is x^2 -2x + 1 = (x-1)^2.
1 is the only eigenvalue. Ker (I - A) = the kernel of
1 1
-1 -1
= solutions of x+y = 0, i.e.
{ -y : y in R}
y
which has basis
-1
1
Since the geometric multiplicity (1) is smaller than
the algebraic multiplicity (2), there is no eigenbasis.
6. The matrix is
2 3
4 5
with char. polynomial x^2 - 7x - 2. By the quadratic formula,
the eigenvalues are (7 +/- sqrt{57})/2 where +/- means
plus or minuis and sqrt{57} means the square of 57.
There will be a one-dimensional eigenspace for each eigenvalue
in this case.
The eigenspace for (-7 + sqrt{57})/2 is the kernel of
(7 + sqrt{57})/2 - 2 -3
-4 (7 + sqrt{57})/2 - 5
or
(3 + sqrt{57})/2 -3
-4 (-3 + sqrt{57})/2
(The second row IS a mutliple of the first.)
Then u =
3
(3 + sqrt{57})/2
is an eigenvector.
Entirely similarly, v =
3
(3 - sqrt{57})/2
is an eigenvector for the eigenvalue (-7 - sqrt{57})/2
and u, v give an eigenbasis.
18. The matrix is
0 0 0 0
0 1 0 1
0 0 0 0
0 0 0 1
The characteristic polynomial is x^2(x - 1)^2, and the eigenvalues
are 0, 1, both with algebraic multiplicity two. The eigenspace
for 0 is the kernel of the original matrix, with has basis
e_1, e_3. The eigenspace for 1 is the kernel of
1 0 0 0
0 0 0 -1
0 0 1 0
0 0 0 0
Since the corresponding equations give x_1 = -x_4 = x_3 = 0,
the kernel is one-dimensional and has basis e_2, the only
eigenvector for the eigenvalue 1. Since the geometric multiplicity
is less than the algebraic multiplicity, there is no eigenbasis.
6.4
22. The matrix is
1 3
-4 10
The characteristic polynomial is x^2 + 11x + 22. The eigenvalues
happen to be real:
(-11 +/- sqrt{121 - 88})/2 = (-11 =/- sqrt{33})/2
24. The matrix is
0 1 0
0 0 1
5 -7 3
The characteristic polynomial is
x -1 0
det 0 x -1 =
-5 7 x-3
x(x(x-3) - (-1)(7)) - (-1)(0(x-3) - (-1)(-5)) =
x(x^2 - 3x + 7) +(0 - 5) = x^3 - 3x^2 + 7x - 5.
By inspection, x = 1 is one root, and we can factor
out x-1: we get (x-1)(x^2 - 2x + 5).
The roots of the quadratic are
(2 +/- sqrt(4-20))/2 or
1 +/- 2i
which, together with 1, give the eigenvalues.
28. Call the two complex conjugate eigenvalues a+bi, a-bi.
Since the real eigenvalue is 2 and the trace and determinant
are 8 and 50, resp. we know that the sum of the eigenvalues
is 8 and the product is 50. This gives 2 + 2a = 8 or a = 3,
and 2(a^2 + b^2) = 50 and so 9 + b^2 = 25, from which
b = 4 or -4. This implies that the complex eigenvalues are
3 +/- 4i.