Post on 22-Dec-2015
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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
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.)
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.)