lec08_09222006 - 10.34 Numerical Methods Applied to...

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10.34, Numerical Methods Applied to Chemical Engineering Professor William H. Green Lecture #8: Constructing And Using The Eigenvector Basis. Homework 1) For those who haven’t programmed before – expect it to take time 2) If you get stuck and are beyond the point of learning, stop and move on. The homework is a learning activity. Matrix Definitions A ·w 3 2 1 w w w # # # # # # # # # 3 2 1 λ λ λ 0 0 0 0 0 0 i = λ i ·w i A ·W = W · Λ eigenvalue an eigenvector of A of A symmetric: come from second derivatives of scalars i.e. Hessians j i ij x x V H = 2 are always symmetrical all real symmetric matrices are ‘normal’ transpose (A T ) * = A H (Hermitian conjugate) of a complex- conjugate Square matrices (NxN) if A = A T ‘symmetric’ A = URU H U : unitary if A = A H ‘Hermitian’ upper triangular (R ) if A ·A H = A H ·A ‘normal’ Schur decomposition : schur(A ) if A T = A -1 ‘orthogonal’ A could be dense matrix if A H = A -1 ‘unitary’ U has hermitian conjugate as inverse If a real matrix is symmetric, it is also Hermitian. For normal matrices A = W · Λ ·W H diagonal eigenvectors & unitary A ·W = W · Λ (W H ·W ) Back to eigenvalue problem Hermitian matrices come up in quantum mechanics. All steady states in quantum mechanics are hermitian eigenvalue problems. Unitary matrices also come up in quantum mechanics and are basis transformations.
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