1. Digital Control Systems Vector-Matrix Analysis

2. Definitions

3. Determinants

4. Inversion of Matrices Nonsingular matrix and Singular matrix

5. Inversion of Matrices Finding the Inverse of a Matrix

6. Vectors and Vector Analysis Linear Dependence and Independence of Vectors Necessary and Sufficient Conditions for Linear Independence of Vectors

7. Vectors and Vector Analysis Linear Dependence and Independence of Vectors Necessary and Sufficient Conditions for Linear Independence of Vectors

8. Eigenvalues, Eigenvectors and Similarity Transformation Rank of a Matrix Properties of rank of a matrix

9. Eigenvalues, Eigenvectors and Similarity Transformation Properties of rank of a matrix (cntd.)

10. Eigenvalues, Eigenvectors and Similarity Transformation Eigenvalues of a Square Matrix :

11. Eigenvalues, Eigenvectors and Similarity Transformation Eigenvectors of nxn Matrix Similar Matrices

12. Eigenvalues, Eigenvectors and Similarity Transformation Diagonalization of Matrices If an nxn matrix A has n distinct eigenvalues, then there are n linearly independent eigenvectors. A can be diagonalized by similarity transformation. If matrix Ahas multiple eigenvalue of multiplicity A, then there are at least one and not more than k linearly independent eigenvectors associated with this eigenvalue. A can not be diagonalized but can be transformed to Jordan canonical form. Jordan Canonical Form

13. Eigenvalues, Eigenvectors and Similarity Transformation Jordan Canonical Form (cntd.) Example:

14. Eigenvalues, Eigenvectors and Similarity Transformation Jordan Canonical Form (cntd.) There exists only one linearly independent eigenvector Two linearly independent eigenvector Three linearly independent eigenvector

15. Eigenvalues, Eigenvectors and Similarity Transformation Similarity Transformation when an nxn matrix has distinct eigenvalues

16. Eigenvalues, Eigenvectors and Similarity Transformation Similarity Transformation when an nxn matrix has multiple eigenvalues = s=1 rank(λI-A)=n-1

17. Eigenvalues, Eigenvectors and Similarity Transformation Similarity Transformation when an nxn matrix has multiple eigenvalues s=1 rank(λI-A)=n-1 (cntd.)

18. Eigenvalues, Eigenvectors and Similarity Transformation Similarity Transformation when an nxn matrix has multiple eigenvalues

19. Eigenvalues, Eigenvectors and Similarity Transformation Similarity Transformation when an nxn matrix has multiple eigenvalues

20. Eigenvalues, Eigenvectors and Similarity Transformation Similarity Transformation when an nxn matrix has multiple eigenvalues n≥s≥2 rank(λI-A)=n-s (cntd.)

21. Eigenvalues, Eigenvectors and Similarity Transformation Similarity Transformation when an nxn matrix has multiple eigenvalues n≥s≥2 rank(λI-A)=n-s (cntd.)

22. Eigenvalues, Eigenvectors and Similarity Transformation Example:

23. Eigenvalues, Eigenvectors and Similarity Transformation Example: rank( )=2

24. Eigenvalues, Eigenvectors and Similarity Transformation Example: :

25. Eigenvalues, Eigenvectors and Similarity Transformation Example: :

26. Eigenvalues, Eigenvectors and Similarity Transformation Example: