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 wi i wi eigenvalue of A A W W an eigenvector of A w1 1 0 0 w2 w3 0 2 0 0 0 3 symmetric come from second derivatives of scalars i e Hessians H ij 2V are always symmetrical xi x j all real symmetric matrices are normal transpose AT AH Hermitian conjugate of a complexconjugate Square matrices NxN A URUH U unitary if A AT symmetric if A AH Hermitian upper triangular R if A AH AH A normal T 1 if A A Schur decomposition schur A orthogonal A could be dense matrix if AH A 1 unitary U has hermitian conjugate as inverse If a real matrix is symmetric it is also Hermitian For normal matrices A W WH diagonal A W W WH W eigenvectors unitary 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 Cite as William Green Jr course materials for 10 34 Numerical Methods Applied to Chemical Engineering Fall 2006 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY 2V Always symmetric because of the equality of mixed partials x i x j Hessian matrix H ij Because they are symmetric they are also normal Similarity Transform identity matrix B S 1wi S 1A S S 1 w S 1A w S 1 iwi i S 1wi A wi iwi B S 1A S B is similar to A A B have the same eigenvalues This is used in practice to calculate eigenvalues Find a diagonal matrix similar to A to find eigenvalues of A S2 1 S1 1A S1 S2 continue to add S and S 1 on each side and eventually you will get at the eingenvalues How to find S if you re GOOD find perfect S such that S A S very difficult to find this S unless someone tells you the eigenvector In quantum mechanics people use matrices of 109 x 109 You have to be very crafty to find the S of such a matrix B Q 1A Q A Q R orthogonal Q 1 T Q B is similar to A 1 Q Q R Q R Q B upper triangular A c wi i c wi QR Algorithm is found in textbook and is very complex does not matter how you scale still get the same eigenvalue eig A gives eigenvalues vectors see help eig Uses EISPACK which is available from netlib Why is this useful singular matrix i 0 cond A max min trace tr A sum of i i aii Initial Conditions Example Problem dy dt A y y t 0 y0 if A is normal dy dt W WHy dt WHy WHy multiply both sides by WH d q t WHy t d q0 WHy0 dt q2 2q2 dq WH dt q dy dt WHW Why d dt q1 1q1 q1 q1 0e 1t look at initial conditions Schur Decomposition y t W q t 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 8 Page 2 of 4 Cite as William Green Jr course materials for 10 34 Numerical Methods Applied to Chemical Engineering Fall 2006 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY q o1e 1t y t W q o 2 e 2t Using eig function you can get W Sometimes things are asymmetrical so eig function will give you a matrix However you can always do Schur decomposition A URUH If you do Schur you get d dt WHy WHy If A were not normal use Schur A URUH dy dt U R UHy d dt UHy Rq q dq dt Rq q qlast t qo laste t dq N 1 0 0 0 R 0 0 0 R N 1 N 1 q N 1 R N 1 N q N t dt dq N 1 R N 1 N 1 q N 1 R N 1 N q N 0 e t1 dt can get this if you weren t sleeping in ODE class this makes solution more difficult than EIG solution Quantum chemistry Something more complicated H x E x interaction between fundamental particles Q e Ei kbT eigenvalues of equation thermo Crafty Solution H ci i x E ci i x x ci i x find these values that will solve H E integrate and multiply by n H c x E c x c H E c n i i n i Property of orthonormal basis functions i dx j ij n Hni i i i i n i ni ci n H i E ci n i H ni ci Ecn H c Ec Eigenvalue Problem 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 8 Page 3 of 4 Cite as William Green Jr course materials for 10 34 Numerical Methods Applied to Chemical Engineering Fall 2006 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY Find the eigenvalues E These are needed for calculations of G free energy thermodynamic constants rate constants and spectroscopic values 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 8 Page 4 of 4 Cite as William Green Jr course materials for 10 34 Numerical Methods Applied to Chemical Engineering Fall 2006 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY
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