Digital Control Systems Vector-Matrix Analysis

Definitions

Determinants

Inversion of Matrices Nonsingular matrix and Singular matrix

Inversion of Matrices Finding the Inverse of a Matrix

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Eigenvalues, Eigenvectors and Similarity Transformation Example:

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

Eigenvalues, Eigenvectors and Similarity Transformation Example: :

Eigenvalues, Eigenvectors and Similarity Transformation Example: :

Eigenvalues, Eigenvectors and Similarity Transformation Example: