Research School of Computer Science Assignment 5 Theory Questions
COMP3670: Introduction to Machine Learning
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 λn.
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 for any matrix B ∈ Rn×n, there can be at most n distinct
eigenvalues for B.
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 = bTa.)
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 R2.
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.