University:
The University Of Western Ontario
Course:
MATH 3020A | Introduction to Abstract Algebra
Academic year:

2024
Views:

216

Pages:

3
Author:

Kyuwon J.

Introduction to Eigenvalues and Eigenvectors Given a square matrix A of size n × n, an eigenvector is a non-zero vector v such that: Av = λv where: • v is the eigenvector, • λ is the corresponding eigenvalue. Key Definitions • Eigenvector: A vector v ̸= 0 such that Av = λv. • Eigenvalue: A scalar λ such that the equation Av = λv has a non-trivial solution. Geometric Interpretation Eigenvectors represent directions in which the transformation described by A scales vectors by the eigenvalue λ. Eigenvectors remain in the same direction (though possibly flipped or scaled). — The Characteristic Equation To find eigenvalues, solve the equation: det(A − λI) = 0 where: • A is the n × n matrix, • λ is a scalar (the eigenvalue), • I is the identity matrix of size n × n, • det denotes the determinant. This determinant equation det(A − λI) = 0 is known as the characteristic equation. — The Characteristic Polynomial The characteristic equation expands into a polynomial in λ, called the characteristic polynomial: p(λ) = det(A − λI) • p(λ) is a polynomial of degree n, • The roots of p(λ) = 0 are the eigenvalues of A. Example: 2x2 Matrix Let: A= a c b d The characteristic polynomial is: det(A − λI) = det 1 a−λ c b d−λ Expanding the determinant: p(λ) = (a − λ)(d − λ) − bc p(λ) = λ2 − (a + d)λ + (ad − bc) where: • a + d is the trace of A, • ad − bc is the determinant of A. — Steps to Compute Eigenvalues and Eigenvectors Step 1: Compute the Characteristic Polynomial 1. Compute A − λI. 2. Find det(A − λI). Step 2: Solve for Eigenvalues Solve p(λ) = 0 to find the eigenvalues λ1 , λ2 , . . . , λn . Step 3: Find Eigenvectors For each eigenvalue λi : 1. Solve the equation: (A − λi I)v = 0 2. Find the non-trivial solutions for v. — Example: Eigenvalues and Eigenvectors of a 3x3 Matrix Let:  2 A = 1 0  0 1 2 1 2 1 Step 1: Compute the Characteristic Polynomial   2−λ 1 0 2−λ 1  A − λI =  1 0 1 2−λ The determinant is:  2−λ det(A − λI) = det  1 0  1 0 2−λ 1  1 2−λ Expanding along the first row: 2−λ det(A − λI) = (2 − λ) det 1 1 1 − 1 · det 2−λ 0 = (2 − λ)((2 − λ)2 − 1) − (1)(2 − λ) = (2 − λ)(λ2 − 4λ + 3) − 2 + λ = λ3 − 6λ2 + 11λ − 6 2 1 2−λ Step 2: Solve for Eigenvalues Solve: λ3 − 6λ2 + 11λ − 6 = 0 Factoring: (λ − 1)(λ − 2)(λ − 3) = 0 The eigenvalues are: λ1 = 1, λ2 = 2, λ3 = 3 Step 3: Find Eigenvectors For each eigenvalue λi , solve (A − λi I)v = 0. For example:  1 A − λ1 I = 1 0 For λ1 = 1 : Solve:  1 1 0 1 1 1     0 x 0 1 y  = 0 1 z 0 The eigenvector corresponding to λ1 = 1 is:  1 v1 = −1 0  (Perform similar steps for λ2 = 2 and λ3 = 3). 3 1 1 1  0 1 1