The Austalian National University Semester 2, 2020
Research School of Computer Science Assignment 5 Theory Questions
Liang Zheng
COMP3670: Introduction to Machine Learning
Question 1 Properties of Eigenvalues
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.
3. Hence, or otherwise, prove that
det(A−1) =
1
detA
You may not use the property det(AB) = det(A) det(B) for this question without proving it.1
Question 2 Properties of Eigenvalues II
1. Let B be a square matrix. Let λ be an eigenvalue of B.
Prove that for all integers n ≥ 1, λn is an eigenvalue of Bn.
2. Let B be a square matrix. Prove that B and BT have the same set of eigenvalues.
Question 3 Properties of Determinants
1. Let U be an square n × n upper triangular matrix. Prove that the determinant of U is equal
to the product of the diagonal elements of U.
2. Let U be an square n× n lower triangular matrix. Prove that the determinant of U is equal to
the product of the diagonal elements of U.
(Hint: Use the previous exercise to help you.)
Question 4 Eigenvalues of symmetric matrices
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 5 Similar Matrices
Let A and B be square matrices. Assume that A is similar to B.
1. Prove that B is similar to A.
2. Prove that A and B share the same characteristic polynomial. (Hint: Note that I = PP−1). You
may use that property that det(AB) = det(A) det(B).
1The question is trivial with this property, and can be proven without this property.
1
Question 6 Computations with Eigenvalues
Let A =
[
2 5
3 4
]
.
1. Compute the eigenvalues of A.
2. Find the eigenspace Eλ for each eigenvalue λ.
3. Verify the eigenspectra spans R2.
4. Hence, find an invertable matrix P and a diagonal matrix D such that A = PDP−1.
5. Hence, or otherwise, find a closed form formula for An for any integer n ≥ 0.
2

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