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review3_2009_10_09

# review3_2009_10_09 - 1 EE263 Review session 3 Jong-Han Kim...

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1 EE263 Review session 3 Jong-Han Kim 2009.10.09. EE263 Review session 3 Eigenvalues / eigenvectors Symmetric matrices Quadratic forms Positive definiteness Examples

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2 EE263 Review session 3 Jong-Han Kim 2009.10.09. Eigenvalues and eigenvectors For A R n × n , there exists nonzero v C n such that Av = λv any such v is called an eigenvector of A associated with eigenvalue λ how to compute ( λI A ) has nonzero null space, thus det ( λI A ) = 0 which is a polynomial of order n the following matlab command finds eigenvectors and eigenvalues [V D] = eig(A);
3 EE263 Review session 3 Jong-Han Kim 2009.10.09. examples consider A = bracketleftbigg 1 2 2 1 bracketrightbigg λ makes ( λI A ) rank deficient, λ 1 = 3 and λ 2 = 1 v null ( λI A ) , v 1 = [1 1] T and v 2 = [1 1] T what if A is rank-deficient already? for example, A = bracketleftbigg 1 2 2 4 bracketrightbigg

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4 EE263 Review session 3 Jong-Han Kim 2009.10.09. examples what are the eigenvalues of A = 1 2 3 0 4 5 0 0 6 ? what are the eigenvalues of A = 1 0 0 0 2 0 0 0 3 ? what are the eigenvalues of A = 1 0 5 2 4 9 3 3 1 5 0 0 4 0 0 0 0 5 7 0 0 0 9 8 2 ?
5 EE263 Review session 3 Jong-Han Kim 2009.10.09. Stability of linear dynamical systems a discrete time linear dynamical system x ( k + 1) = Ax ( k ) + Bu ( k ) y ( k ) = Cx ( k ) + Du ( k ) is stable if and only if max | λ ( A ) | < 1 a continuous time linear dynamical system ˙

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