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EP220 Lecture notes - prepared by- Prof. Dr. Eser OLĞAR 3.3 Power of a Matrix Example: 18 2014-Fall EP220 Lecture notes - prepared by- Prof. Dr. Eser OLĞAR Example: 3.4 Application of eigenvalues to solving systems of Differential Equations In this section we shall apply the matrix analysis, eigenvalues, and eigenvectors to solve systems of first order ordinary differential equations. Differential equations and their systems arise in many areas of mathematics and engineering, for examples in control theory and in analysis of electrical circuits. The unknows in this equations are functions. A number of techniques have been developed to solva such systems of equations. The Laplace transform is one example of this techniques. We shall study anather analytical technique based on eigenvalues and eigenvectors. 18 2014-Fall EP220 Lecture notes - prepared by- Prof. Dr. Eser OLĞAR Example: 18 2014-Fall EP220 Lecture notes - prepared by- Prof. Dr. Eser OLĞAR 2014-Fall Example: Solve the homogeneous linear system using eigenvalues and eigenvectors. 18 EP220 Lecture notes - prepared by- Prof. Dr. Eser OLĞAR 2014-Fall 2. Show that the matrix is not digonalizable. 3. Show that the matrix is digonalizable. 4. Show that the matrix is digonalizable with only one eigenvalue. 5. Find P and D such that P-1AP =D where 6. Find P and D such that P-1AP =D 7. Solve the following differential equations using eigenvalues Exercises 1. Produce a matrix that diagonalizes the given matrix, or show that this matrix is not diagonalizable. 0 −1 5 3 1 0 −5 3 a) b) c) d) 4 3 −4 1 1 3 0 9 0 0 0 −2 0 0 2 0 0 5 0 0 e) 1 0 3 f) 1 0 2 g) 0 h) 0 2 1 2 1 0 1 3 0 −1 2 0 −1 2 0 0 −2 18