MA 35100 Lecture Notes - Augmented Matrix, Fot, If And Only If
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Roger eoel strategyto find eigenvectors and eigenvalues for a. For those where det (a-xi), = 0 characteristic ea. of a polynomial in a called the characteristic polynomial of a-pa(x) - agree is n. Thm 5. 3: if a is nyn, then x is an eigenvalue of a iff det (a-x i) = 0. *a= ("29] has eigenvalues given by its characteristic equation det (a-xi) = 0. To find the eigenvectors, now solve ax= xx i. e. (a-xix: o for each eigenvalue: x= - 3: solve ca - 1-311) x = 0 > (a + 31) x = 0 (29]-(49]=(9] (29] x =0 writeanaugmented matrixis. Solutions to (a + 11x = 0 i. e. nullspace (a +31) = span[[-2, 1753 i. e. (1. 73 is a basis for the -3 eigenspace. For a x= 5: solve (a-51) x = & ce]. x = 0. If ax = xx(x10), then a2x=a(xx)=x(ax) = x
