Complex Eigenvalues kshum ENGG2420B

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 ENGG2420B

Example: flip A linear transformation L(x,y) given by: L(x,y) = (x, -y) x  x y  – y kshum ENGG2420B

Example: shear action A linear transformation given by L(x,y) = (x+0.25y, y) x  x+ 0.25 y y  y kshum ENGG2420B

Repeated eigenvalues, one linearly independent eigenvector What are the eigenvalues of ? Eigenvectors ? Solve \det \begin{bmatrix}1-\lambda& c\\0 & 1-\lambda \end{bmatrix} = 0 ( k nonzero ) kshum ENGG2420B

Example: Expansion L(x,y) = (ax, ay), for some constant a. x  ax y  ay kshum ENGG2420B

Repeated eigenvalues, two linearly independent eigenvectors What are the eigenvalues of ? Eigenvectors ? Solve \det \begin{bmatrix}1-\lambda& c\\0 & 1-\lambda \end{bmatrix} = 0 All non-zero vectors are eigenvector. kshum ENGG2420B

Example: Rotation Rotation by 90 degrees counter-clockwise: L(x,y) = (– y , x). x – y y  x kshum ENGG2420B

Eigenvalues = ? No real root \det \begin{bmatrix}1-\lambda& c\\0 & 1-\lambda \end{bmatrix} = 0 kshum ENGG2420B

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 ENGG2420B

Complex Eigenvalues \det \begin{bmatrix}1-\lambda& c\\0 & 1-\lambda \end{bmatrix} = 0 kshum ENGG2420B