class_4 - CS205 Class 4 1 As a review all the matrices A we...

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CS205 - Class 4 1. As a review, all the matrices A we have looked at up to this point in the class have been full rank. a. For matrices with full rank, the first thing to consider is whether or not it is square. i. If the matrix is square, it is invertible, and Gaussian Elimination can be used to get an LU factorization. Furthermore, if the matrix is symmetric positive definite, a faster Cholesky factorization can be done to get LL T . ii. If the matrix is not square, then it is taller than it is wide, and in this case we do the QR factorization to get the solution. We also considered using the normal equations instead of QR, but said this was bad since it squares the condition number. For QR, there are two ways we consider doing it: Gram-Schmidt and Householder. G.S. has numerical drift for larger matrices, so we prefer to use Householder. b. We will next consider matrices without full rank. In considering these types of matrices, we will look at the Singular Value Decomposition and Principal Component Analysis. In order to talk about these methods we first review eigenvalues and eigenvectors. Eigenvalues and Eigenvectors (Readings Heath pp157-160) 2. For an n n × matrix A, Ax x λ = is the standard eigenvalue problem where λ is an eigenvalue and x is a left eigenvector. a. The right eigenvectors y are defined by T T y A y λ = . If y is a right eigenvector of A, then it is a left eigenvector of T A , since T A y y λ = . b. Usually we refer to “left” eigenvectors simply as eigenvectors while still referring to “right” eigenvectors as right eigenvectors. 3. Complex matrices a. Hermitian matrices have H A A = where the “H” superscript indicates the complex conjugate of the transpose. Thus, for matrices with real values only, H T A A = , i.e. “H” corresponds to simple transposition b. Unitary matrices have H H A A AA I = = . c. Normal matrices have H H A A AA = 4. Recall an eigenvalue is a scalar that satisfies Ax x λ = . a. The set of all eigenvalues of A is called the spectrum of A, and the spectral radius of A is the magnitude of its largest magnitude eigenvalue. ( ) max i i A ρ λ = . b. An eigenvector is a direction along which the action of a matrix is rather simple. The matrix merely expands or contracts vectors in that direction (according to the eigenvalue) leaving the direction unchanged. c. For diagonal matrices, the eigenvalues are on the diagonal, and the eigenvectors are the columns of the identity matrix. For example, 2 0 1 1 2 0 3 0 0      =           and 2 0 0 0 3 0 3 1 1      =     
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