MATH 2270 Lecture Notes - Generalized Eigenvector, Augmented Matrix, Free Variables And Bound Variables
21 Dec 2013
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6. 4 examples for systems of first-order homogeneous lin- Find the general solution to the system (cid:32) (cid:33) x(cid:48) = First, we need to nd the eigenvalues of the matrix of coe cients in this system. The eigenvalues are given by the characteristic equation: (3 )( 1 ) 12 = 0. 3 2 + 2 12 = 0. Thus, we obtain 1 = 5 and 2 = 3. Now, nding eigenvectors, we rst consider 1, which was 5. Substituting this into the (a - i 1)x = 0, we obtain (in augmented matrix form): which reduces to (cid:33) 12 6 0 (cid:32) 2 (cid:32) 2 1 0 (cid:33) One row of zeroes gives us one free variable to work with, and thus one eigenvector. Indeed, we nd that if x2 = t, then 2x1 + t = 0, or x1 = eigenvector. Now, consider the other eigenvalue, 2 = 3. Subbing into (a - i 1)x = 0 and.
