Matrix Definitions Schur Decomposition Quantum chemistry10.34, 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·w321www⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛#########⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛321λλλ000000i = λi·wi A·W = W·Λ eigenvalue an eigenvector of A of A symmetric: come from second derivatives of scalars i.e. Hessians jiijxxVH∂∂∂=2 are always symmetrical all real symmetric matrices are ‘normal’ transpose (AT)* = AH (Hermitian conjugate) of a complex- conjugate Square matrices (NxN) if A = AT ‘symmetric’ A = URUH U: unitary if A = AH ‘Hermitian’ upper triangular (R) if A·AH = AH·A ‘normal’ Schur decomposition: schur(A) if AT = A-1 ‘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·Λ·WH diagonal eigenvectors & unitary A·W = W·Λ(WH·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. 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].Hessian matrix: jiijxxVH∂∂∂=2. Always symmetric, because of the equality of mixed partials. Because they are symmetric, they are also ‘normal’. Similarity Transform identity matrix A·wi = λiwi B(S-1wi) = S-1A(S·S-1)w = S-1A·w = S-1λiwi = λi(S-1wi) 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 Æ Æ Λ How to find S? continue to add S and S-1 if you’re GOOD – find perfect S such that S·A·S = Λ on each side and eventually very difficult to find this S unless someone tells you the you will get at the eingenvalues eigenvector 10.34, Numerical Methods Applied to Chemical Engineering Lecture 8 Prof. William Green 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]. In quantum mechanics people use matrices of 109 x 109“You have to be very crafty to find the S of such a matrix.” A = Q·R B = Q-1A·Q (B is similar to A) orthogonal upper = (Q-1Q)R·Q = R·Q = B (QR Algorithm is found in textbook (Q-1 = QT) triangular and is very complex) A(c·wi) = λi(c·wi) 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 Æ λi = 0 cond(A) = |λ|max/|λ|min trace: tr(A) – sum of λi (Σ(λi) = Σaii) Example Problem Initial Conditions dy/dt = A·y y(t=0) = y0if A is normal dy/dt = WΛWHy multiply both sides by WH WH dy/dt = WHWΛWhy d/dt(WHy) = Λ(WHy) q(t) = WHy(t) dq/dt = Λq d/dt(q1) = λ1q1 q1 = q1,0eλ1t d/dt(q2) = λ2q2 q0 = WHy0 look at initial conditions Schur Decomposition y(t) = W·q(t)⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=#totoeqeqWty2121)(λλ 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 = URUHIf you do Schur you get: d/dt(WHy) = λ(WHy) If A were not normal, use Schur: A = URUHdy/dt = U·R10.34, Numerical Methods Applied to Chemical Engineering Lecture 8 Prof. William Green 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]. (UHy) d/dt(UHy) = Rq dq/dt = Rq (q) q qlast(t) = qo,lasteλt10,111,11,111,11)(tNNNNNNNNNNNNNNeqRqRdtdqtqRqRdtdqλ−−−−−−−−−−+=+= ⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛=λ000000"""""R can get this if you weren’t sleeping in ODE class (this makes solution more difficult than EIG solution) Quantum chemistry Something more complicated ∑−(Ψ=(ΨTkEbieQxExH/~))ˆ interaction between eigenvalues (thermo) fundamental particles of equation Crafty Solution Ψ(x)=Σciφi(x) ∑∑= )()(ˆxcExcHiiiiφφ find these values that will solve ψψEH= integrate and multiply by φn*()()∫∑∫∑∫∑∫∑==iniiniiiniincEHcxcExcHφφφφφφφφ****ˆ)()(ˆ Property of orthonormal basis functions: Hijijdx∫=δφφ*ni δni Problem} e{Eigenvalu ˆ**cEcHEccHcEHcniniiniini===∑∫∑∫∑φφφφ10.34, Numerical Methods Applied to Chemical Engineering Lecture 8 Prof. William Green 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]. Find the eigenvalues E. These are needed for calculations of G (free energy), thermodynamic constants, rate constants, and spectroscopic
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