Graded Assignment 7 (module 13)
Eigenvalues and eigenvectors
Problem 1: For the matrix
5 4 
A

1 2 
(a) Find the eigenvalues of A .
(b) Find the eigenspace associated with each eigenvalue. Give a basis for the space, and state the
dimension of the space.
Do this one completely by hand – all values work out nicely and you get a factorable quadratic at the
end.
Problem 2: For the matrix
5 2 2
A   4 3 4
 4 6 7 
(a) Find all eigenvalues of A . They are all integers - numbers should work out nicely.
(b) For each eigenvalue, find a basis for the associated eigenspace, and express the space as a
generated space of linear combinations of eigenvectors of A (i.e., exactly as in the examples).
Give the dimension of each space.
Please do this one by hand, as much as possible – show the expanded determinant and characteristic
polynomial (but you may need a solver to solve it), and show the matrices being set up for the
eigenvectors (but you can use Scilab to rref them). In other words, don’t just type “spec(A)” into SciLab,
but work the problem in the manner that the suggested problems are shown.
Problem 3: For the matrix
 1 3 4
A   4 5 1 
 3 3 2
use SciLab to find the eigenvalues of A (some of which are complex), and give a basis for the eigenspace
associated with each eigenvalue.
Problem 4:
Answer the following questions:
(a) It is known that a 3x3 matrix A has eigenvalues 1  5 , 2  4 , and 3  10 .
i.
ii.
Is A invertible and how do you know? What are the eigenvalues of A1 ?
What must the algebraic and geometric multiplicities of each of the eigenvalues be?
How do you know?
iii.
What are the eigenvalues of the matrix B  3 A2  6 A  I ?
(b) Construct a 3x3 complex valued matrix that is guaranteed to have real eigenvalues, and explain
how you know this.
Problem 5:
For each of the matrices given below, state the eigenvalues of the matrix, and give the algebraic and
geometric multiplicities of each. For each eigenvalue, give a basis for the eigenspace. OK to do this one
completely in Scilab, but be sure to interpret the answers it gives you (show tiny numbers like 1D-16 as
zeros, and be sure to discard duplicate basis vectors).
(a)
7
0

A  0

0
1
0 0 0
7 0 0
1 6 0
0 0 6
0 0 0
0
0 
0

0
1 
(b)
 3
0

0
A
0
0

0
0 0
7 1
0 7
0 0
0 0
0 0
0
0 0 0 
1 0 0

6 0 0 
0 6 0 

0 0 6 
0
0
Problem 6: In the material on singular values and matrix norms, we noted that while the determinant of
a matrix is a function that takes a matrix and returns a scalar (and as such seems to be a likely candidate
for a norm), it fails most of the norm properties. Show this.
For the space of 2  2 matrices, M 22 , define a function on a matrix A in M 22 by
|| A ||| det( A) |
and run it through the four norm properties, indicating which it passes and which it fails. Provide
counterexamples for the properties that it fails. This one is sketched out / partially done in the lecture,
so the main thing you need to do is provide the specific counterexamples.
Problem 7: For the matrix shown below, compute the eigenvalues (quickly, use Scilab) and give the
spectral radius  ( A) .
 3 1
 4
1
A
 5 2

 10 5
2 1
1 1 
4 8

1 7
Problem 8: For the matrix shown below, compute the singular values (quickly, use Scilab) and give the
norm of A .
 3 1
 4
1
A
 5 2

 10 5
2 1
1 1 
4 8

1 7