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Lecture

10 matrices and eigensvalues, eigenfunctions.pdf
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Department
Chemistry
Course
CHEM 5
Professor
Douglas Tobias
Semester
Fall

Description
LECTURE 11-25-13 MATRICES Matrices: a rectangular array of numbers that ovey certain rules of algebra. An m x n matrix consists of m rows and n columns. Example 1: 2 x 3 matrix: a = 88-2, 1, 3, :l Ø I5 + 33 M>> 2 2 We can skip the secular equation by using the Eigenvalues command: [email protected]; l = [email protected] 1 1 : I5 + 33 M, I5 - 33 M> 2 2 The eigenvectors may be obtained using the Eigenvectors command: [email protected] 1 1 :: I-3 + 33 M, 1>, : I-3 - 33 M, 1>> 6 6 There is another command, Eigensystem, that gives the eigenvalues and eigenvectors all in one shot: [email protected] 1 1 1 1 :: I5 + 33 M, I5 - 33 M>, :: I-3 + 33 M, 1>, : I-3 - 33 M, 1>>> 2 2 6 6 Now let’s verify the similarity transform. First, we store the eigenvectors as columns in the matrix x NOTE: they were given as rows, so we need to transpose the eigenvectors matrix returned by the Eigenvector command Printed by Wolfram Mathematica Student Edition 8 10 - matrices and eigens.nb x = [email protected]@aDD; [email protected] 1 1 6 I-3 + 33 M 6I-3 - 33 M 1 1 Here is the similarity transform: xinv = [email protected]; xinv.a.x; [email protected]%D êê MatrixForm -16 5.37228 8.88178 µ 10 -17 4.16334 µ 10 -0.372281 Use Chop command to zero out the numbers that are effectively zero: [email protected]%D êê MatrixForm 5.37228 0 K O 0 -0.372281 Compare to the diagonal matrix of eigenvalues: [email protected]; [email protected]%D êê MatrixForm K 5.37228 0. O 0. -0.372281 IT WORKS!! Now we do another example involving a 3 x 3 matrix: [email protected]; a = 881, 2, 3 r r The general form of the volume element in a coordinate system defined by the coordinates u , u , u is 1 2 3 dV = h1h 2 3u d1 du2, w3ere h , 1 , 2 a3e “scale factors” that are particular to a given coordinate system. For example: In the Cartesian coordinate system, the coordinates are1u = x,2u = y,3u = z, and the scale factors are: Printed by Wolfram Mathematica Student Edition 10 10 - matrices and eigens.nb [email protected]; 8h1, h2, h3< = [email protected]"Cartesian", "ScaleFactors", 8x, y, z
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