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

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Eigenvalues, Eigenvectors and Similarity Transformation

Eigenvectors of nxn Matrix

Similar Matrices

Eigenvalues, Eigenvectors and Similarity Transformation

Diagonalization of MatricesIf 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 eigenvaluess=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 eigenvaluesn≥s≥2 rank(λI-A)=n-s (cntd.)

Eigenvalues, Eigenvectors and Similarity Transformation

Similarity Transformation when an nxn matrix has multiple eigenvaluesn≥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: