CS 205A class_5 notes

Scientific Computing

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CS205 – Class 5 Covered in class: 1, 3, 4, 5. Reading: Heath Chapter 4. Eigenvalues / Eigenvectors Continued 1. Idea is to compute eigenvalues using matrix form because it is in general easier than solving a n -degree polynomial. a. Must be careful though as we need to preserve least squares solution. i.e. Gaussian- Elimination will not work. b. Use a similarity transform that produces a new matrix with “same” eigenvalues/eigenvectors. 2. Formally, a matrix A is said to be similar to a matrix B, if 1 B TA T = for a nonsingular matrix T. a. If A and B are similar, then they have the same eigenvalues. B yy λ = or 1 TA T y y = or () () A Ty Ty = . Note that the eigenvectors of A are Ty where y are the eigenvectors of B (this formalizes the notion of sameness from 3(b)). b. If the matrix A has distinct eigenvalues (no repeated eigenvalues), then similarity transforms can be used to put it into diagonal form where the eigenvalues can be read from the diagonal and the eigenvectors are the columns of the identity matrix. Then the eigenvectors of A are T times the columns of the identity matrix, i.e. the columns of T. c. If A is real and symmetric, an orthogonal T can be used to put A into diagonal form. Moreover, the eigenvalues are real valued.
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CS 205A class_5 notes - CS205 Class 5 Covered in class: 1,...

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