1. Eigensystems - IntroJacob Y. Kazakia © 20051 Eigensystems 1

2. Eigensystems - IntroJacob Y. Kazakia © 20052 Eigensystems 2

3. Eigensystems - IntroJacob Y. Kazakia © 20053 Eigensystems 3

4. Eigensystems - IntroJacob Y. Kazakia © 20054 Eigensystems 4

5. Eigensystems - IntroJacob Y. Kazakia © 20055 Eigensystems 5

6. Eigensystems - IntroJacob Y. Kazakia © 20056 Eigensystems 6

7. Eigensystems - IntroJacob Y. Kazakia © 20057 Calculating Determinants Given a nxn matrix A as: its minor A ij is defined as the matrix obtained by eliminating the i th row and j th column. For example the minor A 22 of the matrix is the (n-1)x(n-1) matrix or We define the determinant by the first row expansion here the power of -1 makes the sign alternate from positive to negative

8. Eigensystems - IntroJacob Y. Kazakia © 20058 Calculating Determinants - examples for a 2x2 matrix the determinant calculation is trivial. For example: for a three by three matrix we have Things get more difficult for a 4x4 matrix since, in the expansion we must calculate 4, 3x3 determinants. There are other short cut ways for calculating numerical determinants. MATLAB does this effortlessly.

9. Eigensystems - IntroJacob Y. Kazakia © 20059 Systems of Differential Equations Consider the 3X3 system of first order differential equations: We write it in matrix form as: For each eigenvector of the matrix consequently we can have or equivalently:

10. Eigensystems - IntroJacob Y. Kazakia © 200510 Systems of Differential Equations 2 Here K is the matrix of eigenvectors and D is a diagonal matrix. If we can find 3 linearly independent eigenvectors, then we can construct the inverse of K and hence obtain: This is known as a similarity transformation and provides the means of diagonalizing a given matrix Once we know the eigenvalues and eigenvectors of the coefficient matrix, the solution of the system of differential equations can be explicitly written as: Here c1, c2, c3 are arbitrary coefficients. The derivation of this solution is shown in the next slide

11. Eigensystems - IntroJacob Y. Kazakia © 200511 Systems of Differential Equations 3 In the system use the transformation: We then obtain: This produces trivially the solutions for y’s as: The functions x are then obtained from:

12. 12 S.D.E. 4 - Complete Solution For our matrix we write the characteristic equation: The expansion The standard formThe factorization The determinant