EIGENVALUES AND EIGENVECTORS ELECTRONIC VERSION OF LECTURE Dr Lê Xuân Đại HoChiMinh City University of Technology Faculty of Applied Science, Department of Applied Mathematics Email: ytkadai@hcmut.edu.vn HCMC — 2018 Dr Lê Xuân Đại (HCMUT-OISP) EIGENVALUES AND EIGENVECTORS HCMC — 2018 / 56 OUTLINE THE REAL WORLD PROBLEMS EIGENVALUES AND EIGENVECTORS OF A MATRIX DIAGONALIZATION MATL AB Dr Lê Xuân Đại (HCMUT-OISP) EIGENVALUES AND EIGENVECTORS HCMC — 2018 / 56 The Real World Problems MODELLING MOTION PQR → P Q R is the reflection over the x−axis Dr Lê Xuân Đại (HCMUT-OISP) EIGENVALUES AND EIGENVECTORS HCMC — 2018 / 56 The Real World Problems A= 0 −1 is the reflection matrix Therefore, for every point in the plane (x1 , x2 ), the matrix that results in a reflection over the x−axis and then we obtain a new point in the plane (y1, y2) x1 −x2 Question: For every point (x1, x2), find y1 x1 = Ak , (k ∈ N) y2 x2 y1 y2 = x1 −1 x2 Dr Lê Xuân Đại (HCMUT-OISP) = EIGENVALUES AND EIGENVECTORS HCMC — 2018 / 56 Eigenvalues and Eigenvectors of a Matrix A= Definition −1 , u= ,v= We have −1 −1 A −1 −1 = −1 A 0 = = −1 −1 Dr Lê Xuân Đại (HCMUT-OISP) and EIGENVALUES AND EIGENVECTORS HCMC — 2018 / 56 Eigenvalues and Eigenvectors of a Matrix Definition DEFINITION 2.1 If A is an n × n matrix, then a nonzero vector X ∈ Rn , X = is called an eigenvector of A if AX = λ.X for some scalar λ The scalar λ is called an eigenvalue of A and X is said to be an eigenvector corresponding to λ EXAMPLE 2.1 Find eigenvalues and eigenvectors of A= 0 −1 Dr Lê Xuân Đại (HCMUT-OISP) EIGENVALUES AND EIGENVECTORS HCMC — 2018 / 56 Eigenvalues and Eigenvectors of a Matrix Definition The equation AX = λX can be rewritten as (A − λI)X = x1 −1 x2 = 1−λ 0 −1 − λ λx1 λx2 x1 x2 ⇔ = This homogeneous linear system has non-zero solution X = 0, thus 1−λ = ⇔ λ2 − = 0 −1 − λ ⇔ λ1 = −1, λ2 = Dr Lê Xuân Đại (HCMUT-OISP) EIGENVALUES AND EIGENVECTORS HCMC — 2018 / 56 Eigenvalues and Eigenvectors of a Matrix Definition In the case where λ1 = −1, we have 2x1 + 0x2 = ⇔ x1 = 0, x2 = α 0x1 + 0x2 = Therefore, the eigenvectors corresponding to λ1 = −1 are α(0, 1), α = In the case where λ2 = We have 0x1 + 0x2 = ⇔ x1 = β, x2 = 0x1 − 2x2 = Therefore, the eigenvectors corresponding to λ2 = are β(1, 0), β = Dr Lê Xuân Đại (HCMUT-OISP) EIGENVALUES AND EIGENVECTORS HCMC — 2018 / 56 Eigenvalues and Eigenvectors of a Matrix Definition EXAMPLE 2.2 Find eigenvalues and eigenvectors of A= −2 Dr Lê Xuân Đại (HCMUT-OISP) EIGENVALUES AND EIGENVECTORS HCMC — 2018 / 56 Eigenvalues and Eigenvectors of a Matrix Definition AX = λX can be rewritten λx1 λx2 ⇔ 1−λ −2 − λ x1 x2 −2 = x1 x2 = This homogeneous linear system has non-zero solution X = 0, thus 1−λ = ⇔ (1 − λ)2 + = −2 − λ ⇔ λ1,2 = ± 2i Dr Lê Xuân Đại (HCMUT-OISP) EIGENVALUES AND EIGENVECTORS HCMC — 2018 10 / 56 ... corresponding to λ EXAMPLE 2.1 Find eigenvalues and eigenvectors of A= 0 −1 Dr Lê Xuân Đại (HCMUT-OISP) EIGENVALUES AND EIGENVECTORS HCMC — 2018 / 56 Eigenvalues and Eigenvectors of a Matrix Definition... Matrix Definition EXAMPLE 2.2 Find eigenvalues and eigenvectors of A= −2 Dr Lê Xuân Đại (HCMUT-OISP) EIGENVALUES AND EIGENVECTORS HCMC — 2018 / 56 Eigenvalues and Eigenvectors of a Matrix Definition... Dr Lê Xuân Đại (HCMUT-OISP) EIGENVALUES AND EIGENVECTORS HCMC — 2018 12 / 56 Eigenvalues and Eigenvectors of a Matrix Characteristic equation FINDING EIGENVALUES AND EIGENVECTORS OF A SQUARE MATRIX