# 6 some definions eigenvalues and eigenvectors

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Unformatted text preview: igenvalue of A-1 with associated eigenvector xi. The eigenvalues of a diagonal matrix D = diag(d1, . . . dn) are just the diagonal entries d1, . . . dn. Diagonalizable: We can write all the eigenvector equations together as AX = X. If the eigenvectors of A are linearly independent, A = XX-1. We say A is diagonalizable. 8 Eigenvalues and Eigenvectors of Symmetric Matrices A: a symmetric matrix Sn All the eigenvalues of A are real. The eigenvectors of A are orthonormal (The inner product is 0.). A is diagonalizable: A = UUT (Note: U-1 =UT) All i > 0 A is positive definite All i 0 A is positive semidefinite A has both positive and negative eigenvalues A is indefinite 9 What is Matrix Calculus Calculus Differential calculus Derivative e.g. f(x)=x2, derivative function f '(x)=2x Integral calculus Matrix Calculus Extension of calculus to the vector/matrix setting Gradient Hessian 10 The Gradient Definition Function f : Rmn R A: m n matrix The gradient of f (written as Af(A)) is an m n matrix and each element of the matrix is a partial derivative defined by 11 The Gradient Example A: 2x2 matrix f(A)=|A| calculate each element of Af(A) 12 The Gradient Example The gradient of f The general case for f(A)=|A| 13 The Gradient When A is a vector a vector x Rn the gradient of f Two properties x(f(x) + g(x)) = xf(x) + xg(x) For t R, x...
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## This note was uploaded on 01/22/2013 for the course EE 103 taught by Professor Vandenberghe,lieven during the Fall '08 term at UCLA.

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