Complex Eigenvalues kshum ENGG 2420 B 1 Steps

Complex Eigenvalues kshum ENGG 2420 B 1

Steps in calculating eigenvalues and eigenvectors • • Given a matrix M. Find the characteristic polynomial. Find the roots of the characteristic polynomial. For each eigenvalue of M, find the non-zero vectors v such that M v = v. kshum ENGG 2420 B 2

Example: flip • A linear transformation L(x, y) given by: L(x, y) = (x, -y) x x y –y kshum ENGG 2420 B 3

Example: shear action • A linear transformation given by L(x, y) = (x+0. 25 y, y) x x+ 0. 25 y y y kshum ENGG 2420 B 4

Repeated eigenvalues, one linearly independent eigenvector • What are the eigenvalues of ? • Eigenvectors ? – Solve ( k nonzero ) kshum ENGG 2420 B 5

Example: Expansion • L(x, y) = (ax, ay), for some constant a. x ax y ay kshum ENGG 2420 B 6

Repeated eigenvalues, two linearly independent eigenvectors • What are the eigenvalues of ? • Eigenvectors ? – Solve All non-zero vectors are eigenvector. kshum ENGG 2420 B 7

Example: Rotation • Rotation by 90 degrees counter-clockwise: L(x, y) = (– y , x). x – y y x kshum ENGG 2420 B 8

Eigenvalues = ? No real root kshum ENGG 2420 B 9

Extension to complex vectors and matrix • Given a square matrix A, a non-zero vector v is called an eigenvector of A, if we an find a number , which may be complex, such that • This number is the eigenvalue of A corresponding to the eigenvector v. kshum ENGG 2420 B 10

Complex Eigenvalues kshum ENGG 2420 B 11