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Linear algebra: matrix Eigen-value Problems

Linear algebra: matrix Eigen-value Problems. Eng. Hassan S. Migdadi. Part 3. Eigenvalue Problems. Eigenvalues and eigenvectors Vector spaces Linear transformations Matrix diagonalization. The Eigenvalue Problem. Consider a n x n matrix A Vector equation: Ax = l x

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Linear algebra: matrix Eigen-value Problems

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  1. Linear algebra: matrix Eigen-value Problems Eng. Hassan S. Migdadi Part 3

  2. Eigenvalue Problems • Eigenvalues and eigenvectors • Vector spaces • Linear transformations • Matrix diagonalization

  3. The Eigenvalue Problem • Consider a nxn matrix A • Vector equation: Ax = lx • Seek solutions for x and l • l satisfying the equation are the eigenvalues • Eigenvalues can be real and/or imaginary; distinct and/or repeated • x satisfying the equation are the eigenvectors • Nomenclature • The set of all eigenvalues is called the spectrum • Absolute value of an eigenvalue: • The largest of the absolute values of the eigenvalues is called the spectral radius

  4. Determining Eigenvalues • Vector equation • Ax = lx  (A-lI)x = 0 • A-lI is called the characteristic matrix • Non-trivial solutions exist if and only if: • This is called the characteristic equation • Characteristic polynomial • nth-order polynomial in l • Roots are the eigenvalues {l1, l2, …, ln}

  5. Eigenvalue Example • Characteristic matrix • Characteristic equation • Eigenvalues: l1 = -5, l2 = 2

  6. Eigenvalue Properties • Eigenvalues of A and AT are equal • Singular matrix has at least one zero eigenvalue • Eigenvalues of A-1: 1/l1, 1/l2, …, 1/ln • Eigenvalues of diagonal and triangular matrices are equal to the diagonal elements • Trace • Determinant

  7. Determining Eigenvectors • First determine eigenvalues: {l1, l2, …, ln} • Then determine eigenvector corresponding to each eigenvalue: • Eigenvectors determined up to scalar multiple • Distinct eigenvalues • Produce linearly independent eigenvectors • Repeated eigenvalues • Produce linearly dependent eigenvectors • Procedure to determine eigenvectors more complex (see text) • Will demonstrate in Matlab

  8. Eigenvector Example • Eigenvalues • Determine eigenvectors: Ax = lx • Eigenvector for l1 = -5 • Eigenvector for l1 = 2

  9. Matlab Examples >> A=[ 1 2; 3 -4]; >> e=eig(A) e = 2 -5 >> [X,e] = eig(A) X = 0.8944 -0.3162 0.4472 0.9487 e = 2 0 0 -5 >> A=[2 5; 0 2]; >> e=eig(A) e = 2 2 >> [X,e]=eig(A) X = 1.0000 -1.0000 0 0.0000 e = 2 0 0 2

  10. Vector Spaces • Real vector space V • Set of all n-dimensional vectors with real elements • Often denoted Rn • Element of real vector space denoted • Properties of a real vector space • Vector addition • Scalar multiplication

  11. Vector Spaces cont. • Linearly independent vectors • Elements: • Linear combination: • Equation satisfied only for cj= 0 • Basis • n-dimensional vector space V contains exactly n linearly independent vectors • Any n linearly independent vectors form a basis for V • Any element of V can be expressed as a linear combination of the basis vectors • Example: unit basis vectors in R3

  12. Inner Product Spaces • Inner product • Properties of an inner product space • Two vectors with zero inner product are called orthogonal • Relationship to vector norm • Euclidean norm • General norm • Unit vector: ||a|| = 1

  13. Linear Transformation • Properties of a linear operator F • Linear operator example: multiplication by a matrix • Nonlinear operator example: Euclidean norm • Linear transformation • Invertible transformation • Often called a coordinate transformation

  14. Orthogonal Transformations • Orthogonal matrix • A square matrix satisfying: AT = A-1 • Determinant has value +1 or -1 • Eigenvalues are real or complex conjugate pairs with absolute value of unity • A square matrix is orthonormal if: • Orthogonal transformation • y = Ax where A is an orthogonal matrix • Preserves the inner product between any two vectors • The norm is also invariant to orthogonal transformation

  15. Similarity Transformations • Eigenbasis • If a nxn matrix has n distinct eigenvalues, the eigenvectors form a basis for Rn • The eigenvectors of a symmetric matrix form an orthonormal basis for Rn • If a nxn matrix has repeated eigenvalues, the eigenvectors may not form a basis for Rn(see text) • Similar matrices • Two nxn matrices are similar if there exists a nonsingular nxn matrix P such that: • Similar matrices have the same eigenvalues • If x is an eigenvector of A, then y = P-1x is an eigenvector of the similar matrix

  16. Matrix Diagonalization • Assume the nxn matrix A has an eigenbasis • Form the nxn modal matrix X with the eigenvectors of A as column vectors: X= [x1, x2, …, xn] • Then the similar matrix D = X-1AX is diagonal with the eigenvalues of A as the diagonal elements • Companion relation: XDX-1 = A

  17. Matrix Diagonalization Example

  18. Matlab Example >> A=[-1 2 3; 4 -5 6; 7 8 -9]; >> [X,e]=eig(A) X = -0.5250 -0.6019 -0.1182 -0.5918 0.7045 -0.4929 -0.6116 0.3760 0.8620 e = 4.7494 0 0 0 -5.2152 0 0 0 -14.5343 >> D=inv(X)*A*X D = 4.7494 -0.0000 -0.0000 -0.0000 -5.2152 -0.0000 0.0000 -0.0000 -14.5343

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