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: