Chapter 7 Eigenvalue Eigenvector Topics 1 Get to
- Slides: 70
Chapter #7 Eigenvalue & Eigenvector
Topics 1. Get to know: Eigenvalue & Eigenvector 2. Estimation of Eigenvalue & Eigenvector 3. Theorem
Get to know: Eigenvalue & Eigenvector � derived from the German word "eigen“ • means "proper" or "characteristic. " � Eigenvalues and the associated eigenvectors • ‘special’ properties of square matrices (n x n) Eigenvalues • parameterize the dynamical properties of the system (timescales, resonance properties, amplification factors, etc) � Eigenvectors • define the vector coordinates of the normal modes of the system. �
Get to know: Eigenvalue & Eigenvector A: a Linear Transformation Square Matrix (n x n) x: Eigenvector (non-zero vector) of A (not unique) ג : Eigenvalue (Scalar value) of A
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Get to know: Eigenvalue & Eigenvector �Each eigenvector • associated with a particular eigenvalue.
Effect of eigenvalue to its eigenvector
Get to know: Eigenvalue & Eigenvector � The general state of the system can be • expressed as a linear combination of eigenvectors.
Get to know: Eigenvalue & Eigenvector �The beauty of eigenvectors is that • They can be made orthogonal (decoupled from one another). • An orthogonal expansion of the system is possible. • The normal modes can be handled independently
Get to know: Eigenvalue & Eigenvector � The dominant eigenvector of a matrix A • an eigenvector corresponding to the eigenvalue of largest magnitude �(for real numbers, largest absolute value) of that matrix. � Many of the "real world" applications are • primarily interested in the dominant eigenpair. • The method used to find the dominant eigenvector is called the power method.
Get to know: Eigenvalue & Eigenvector � The eigenvalue of smallest magnitude of a matrix • is the same as the inverse (reciprocal) of the dominant eigenvalue of the inverse of the matrix. � If the eigenvalue of smallest magnitude is needed, • the inverse matrix A is often used to solve for its dominant eigenvalue. • This is why the dominant eigenvalue is so important.
Estimation: Eigenvalue & Eigenvector
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Example
Example -4 x 1+2 x 2 = 0 2 x 1 - x 2 = 0
Checking the results =
Example x 1+ 2 x 2 = 0 2 x 1 +4 x 2 = 0
Checking the results =
Eigenspace, Spectrum, and Spectral radius Spectrum ��� A={-1, - 6} ��� spectral radius =|- 6| =6
Gaussian elimination
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Leslie model Example
Face Recognition Example �Eigen Face • Principle of Component Analysis (PCA)
Face Recognition using Eigenface
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Linear independent of Eigenvector =5,
Linear independent of Eigenvector Repeated eigenvalue ������������� eigenvector ������ Rn
D: eigenvalue diagonal matrix
Diagonalization
Power of a Matrix A: square matrix (nxn) n: large - Use eigenvalues & eigenvectors to solve
Power of a Matrix Same direction as eigenvector, just scaling the magnitude Any vector could be represented (transformed) as a linear combination of eigenvectors
Power of a Matrix
Power of a Matrix
Power of a Matrix A linear combination of eigenvectors
Power of a Matrix A linear combination of eigenvectors
Power of a Matrix
Eigen Decomposition Theorem Independent eigenvectors P: Matrix of eigenvectors
Eigen Decomposition Theorem Diagonal Matrix of eigenvalues
Eigen Decomposition Theorem
Eigen Decomposition Theorem
Continued
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