Stat 5101 Lecture Notes - School of Statistics

E.2. EIGENVALUES AND EIGENVECTORS 233andx k =z k‖z k ‖ .It is easily verified that this does produce an orthonormal set, and it is onlyslightly harder to prove that none of the x i are zero because that would implylinear dependence of the y i .E.2 Eigenvalues and EigenvectorsIf A is any matrix, we say that λ is a right eigenvalue corresponding to aright eigenvector x ifAx = λxLeft eigenvalues and eigenvectors are defined analogously with “left multiplication”x ′ A = λx ′ , which is equivalent to A ′ x = λx. So the right eigenvaluesand eigenvectors of A ′ are the left eigenvalues and eigenvectors of A. WhenA is symmetric (A ′ = A), the “left” and “right” concepts are the same andthe adjectives “left” and “right” are unnecessary. Fortunately, this is the mostinteresting case, and the only one in which we will be interested. From now onwe discuss only eigenvalues and eigenvectors of symmetric matrices.There are three important facts about eigenvalues and eigenvectors. Twoelementary and one very deep. Here’s the first (one of the elementary facts).Lemma E.1. Eigenvectors corresponding to distinct eigenvalues are orthogonal.thenThis means that ifAx i = λ i x iλ i ≠ λ j implies x ′ ix j =0.(E.3)Proof. Suppose λ i ≠ λ j , then at least one of the two is not zero, say λ j . Thenx ′ ix j = x′ i Ax jλ j= (Ax i) ′ x jλ j= λ ix ′ i x jλ j= λ iλ j· x ′ ix jand since λ i ≠ λ j the only way this can happen is if x ′ i x j =0.Here’s the second important fact (also elementary).Lemma E.2. Every linear combination of eigenvectors corresponding to thesame eigenvalue is another eigenvector corresponding to that eigenvalue.thenThis means that ifAx i = λx i( k∑) ( k∑)A c i x i = λ c i x ii=1i=1