Chapter 7 Eigenvalue Eigenvector Topics 1 Get to

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Chapter #7 Eigenvalue & Eigenvector

Chapter #7 Eigenvalue & Eigenvector

Topics 1. Get to know: Eigenvalue & Eigenvector 2. Estimation of 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“ •

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

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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Page 334

Get to know: Eigenvalue & Eigenvector �Each eigenvector • associated with a particular eigenvalue.

Get to know: Eigenvalue & Eigenvector �Each eigenvector • associated with a particular eigenvalue.

Effect of eigenvalue to its eigenvector

Effect of eigenvalue to its eigenvector

Get to know: Eigenvalue & Eigenvector � The general state of the system can

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

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

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

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

Estimation: Eigenvalue & Eigenvector

Pages 335 -336 a Continued

Pages 335 -336 a Continued

Example

Example

Example -4 x 1+2 x 2 = 0 2 x 1 - x 2

Example -4 x 1+2 x 2 = 0 2 x 1 - x 2 = 0

Checking the results =

Checking the results =

Example x 1+ 2 x 2 = 0 2 x 1 +4 x 2

Example x 1+ 2 x 2 = 0 2 x 1 +4 x 2 = 0

Checking the results =

Checking the results =

Eigenspace, Spectrum, and Spectral radius Spectrum ��� A={-1, - 6} ��� spectral radius =|-

Eigenspace, Spectrum, and Spectral radius Spectrum ��� A={-1, - 6} ��� spectral radius =|- 6| =6

Gaussian elimination

Gaussian elimination

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Page 337 b

Leslie model Example

Leslie model Example

Face Recognition Example �Eigen Face • Principle of Component Analysis (PCA)

Face Recognition Example �Eigen Face • Principle of Component Analysis (PCA)

Face Recognition using Eigenface

Face Recognition using Eigenface

หา inverse ได

หา inverse ได

Linear independent of Eigenvector =5,

Linear independent of Eigenvector =5,

Linear independent of Eigenvector Repeated eigenvalue ������������� eigenvector ������ Rn

Linear independent of Eigenvector Repeated eigenvalue ������������� eigenvector ������ Rn

D: eigenvalue diagonal matrix

D: eigenvalue diagonal matrix

Diagonalization

Diagonalization

Power of a Matrix A: square matrix (nxn) n: large - Use eigenvalues &

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

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

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 A linear combination of eigenvectors

Power of a Matrix A linear combination of eigenvectors

Power of a Matrix

Power of a Matrix

Eigen Decomposition Theorem Independent eigenvectors P: Matrix of eigenvectors

Eigen Decomposition Theorem Independent eigenvectors P: Matrix of eigenvectors

Eigen Decomposition Theorem Diagonal Matrix of eigenvalues

Eigen Decomposition Theorem Diagonal Matrix of eigenvalues

Eigen Decomposition Theorem

Eigen Decomposition Theorem

Eigen Decomposition Theorem

Eigen Decomposition Theorem

Continued

Continued