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Eigen Vector1 - Tuesday Eigen Vectors x=Ax bt...

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Tuesday, November 03, 2009 Eigen Vectors ** = + x Ax bt = x0 x0 ** Eigenvalues/eigenvectors = Awi λiwi - Eigenvalues/Eige vectors of same or different values form linear independent sectors - Each Eigenvalue λ i has at least one Eigenvector - If m i is the multiplicity of λ i it is positive then λ i has 1, 2, 3… or m i eigenvectors Ex3. =- - → A 230 2 only one eigenvector Ex4. =- - A 200 2 2 eigenvector *If there are as many Eigenvectors as the size of the matrix, then there is a full set of the eigenvector nxn matrix A has n eigenvectors Assume a full set of eigenvectors: = AW1W2 Wn W1W2 Wn λ1000λ2000λn || || (eigenvector matrix)(eigenvalue matrix) In a 2x2 case = = AW1W2 W1W2λ100λ2 a11a12a21a22w11w12w21w22 w11w12w21w22λ100λ2 + + + + = a11w11 a12w21a11w12 a12w22a21w11 a22w21a21w12 a22w22 λ1w11λ2w12λ1w21λ2w22 = = = Aw1 λiwi AW1W2 Wn W1W2 Wnλ1000λ2000λn AW Multiply by W -1 if W is nxn matrix (full set of eigen vector) AWW -1= W -1 A = Λ Back to ** in to top box Define xt = Wzt zt = W -1 x ( t ) 1
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Tuesday, November 03, 2009 = + = + ; } = ddtWz AWz bt Wz AWz bt initial conditions change Wz0 x0 = = ddta ft adfdt ddtA ft Adfdt - = - + - → = + ( ) W 1Wz W 1AWz W 1bt z Λz ξ t || || || I Λ ξ = + ( ) - zit zi0eλit eλit0tξi t' e λit' If b (t)=constant
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