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 (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. 1
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