EIGENVALUE AND EIGENVECTOR Eigenvalue problem Examples in notes
EIGENVALUE AND EIGENVECTOR • Eigenvalue problem (Examples in notes page) : Eigenvalue : Eigenvector • How to solve? [X, Lambda]=eig(A) in Matlab – {x} = {0} is a solution (trivial solution) – In order to have non-trivial solution, the determinant must be zero. – Calculate problem from this equation and calculate from the eigenvalue 1
EIGENVALUE AND EIGENVECTOR • Characteristic equation – The textbook has a solution for [A]3 x 3 case • Eigenvectors – After solving for eigenvalues, substitute each of them to eigenproblem – Since is singular, no unique solution exists – Practice example in the textbook (see also notes page) 2
QUADRATIC FORM • Quadratic form: quadratic function of all components • Matrix notation F=x’*A*x in Matlab • Symmetric part is enough ([B] is not sym) 3
POSITIVE DEFINITE MATRIX • Positive definite • Positive semi-definite How is this related to matrices that are almost singular? (answer in notes page) • Positive definiteness => each column of the matrix is linearly independent = the matrix is invertible = the matrix is not singular = the matrix equation has a unique solution. • Positive definite matrices have all positive eigenvalues. 4
Quiz-like problems • Given the matrix the quadratic form and the vector x’=[0 1 1] calculate • Check if the matrix is positive definite. • Answers in the notes page 5
MAXIMA & MINIMA OF FUNCTIONS • Single Variable f(x) – Taylor series expansion – In order for f to be extremum, – Condition for minimum: – Condition for maximum: 6
MAXIMA & MINIMA OF FUNCTIONS cont. • Multi-Variable f(x) – Taylor series expansion – In order for f to be extremum, Hessian matrix Hij – Condition for minimum: [H] is positive definite – Condition for maximum: [H] is negative definite 7
MINIMUM PRINCIPLE • Function in quadratic form – [A]: stiffness of the structure, {x}: displacement, {b}: applied force – Potential energy – structure is in equilibrium when F has a minimum value • Matrix equation – Solution of the matrix equation minimizes the quadratic form F. 8
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