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Question 1 Properties of Eigenvalues (5+5=10 credits)
Let A be an invertible matrix.
1. Prove that all the eigenvalues of A are non-zero.
2. Prove that for any eigenvalue λ of A, λ
−1
is an eigenvalue of A−1
.
Question 2 Properties of Eigenvalues II (10 credits)
Let B be a square matrix. Let x be an eigenvector of B with eigenvalue λ. Prove that for all integers
n ≥ 1, x is an eigenvector of Bn with eigenvalue λ
n
.
Question 3 Distinct eigenvalues and linear independence (20+5 credits)
Let A be a n × n matrix.
1. Suppose that A has n distinct eigenvalues λ1, . . . , λn, and corresponding non-zero eigenvectors
x1, . . . , xn. Prove that {x1, . . . , xn} is linearly independant.
Hint: You may use without proof the following property: If {y1
, . . . , ym} is linearly dependent
then there exists some p such that 1 ≤ p < m, yp+1 ∈ span{y1
, . . . , yp} and {y1
, . . . , yp} is
linearly independent.
2. Hence, or otherwise, prove that A can have at most n distinct eigenvalues.
Question 4 Properties of Determinants (10+15=25 credits)
1. Prove det(AT
) = det(A).
2. Prove det(In) = 1 where In is the n × n identity matrix.
Question 5 Eigenvalues of symmetric matrices (15 credits)
1. Let A be a symmetric matrix. Let v1 be an eigenvector of A with eigenvalue λ1, and let v2 be an
eigenvector of A with eigenvalue λ2. Assume that λ1 6= λ2. Prove that v1 and v2 are orthogonal.
(Hint: Try proving λ1v
T
1 v2 = λ2v
T
1 v2. Recall the identity a
Tb = b
T a.)
Question 6 Computations with Eigenvalues (3+3+3+3+3=15 credits)
Let A =
−1 2
3 4
.
1. Compute the eigenvalues of A.
2. Find the eigenspace Eλ for each eigenvalue λ. Write your answer as the span of a collection of
vectors.
3. Verify the set of all eigenvectors of A spans R
2
.
4. Hence, find an invertable matrix P and a diagonal matrix D such that A = PDP−1
.
5. Hence, find a formula for efficiently 1
calculating An
for any integer n ≥ 0. Make your formula
as simple as possible.
1That is, a closed form formula for An
as opposed to multiplying A by itself n times over.
1
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