MA 35100 Lecture Notes - Eigenvalues And Eigenvectors, Diagonalizable Matrix
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To find eigenvectors, solve (a-xix= & for each x: 1: (a + 1) x = g. A has as a basis for the -1-eigenspace: 3/f)3-dimension=1. A has as a basis for the 3-eigenspace: [(d]. (73-dimension=2. If x is an e-value of a, the geometric multiplicity of x, written gm(x), is the dimension of the -espace. Note: om does not always equal am, even though it does in the above example. Find the e-values of a and their am and om. A has a basis of the 1-eigenspace: [/c7} aim: 1,so jmc1): 1. Theorem 5. 2: if x is an eigenvalue of an nxumatrix a, then sm(x) 1am (x). Am: #of times & appears as a root in pr(x) off. Note: if am1x)=1, 0m(x):1; no need to find eigenvectors. Goal: understand when a, nxh, is such that the has a basis of. E. g. so we can multiply by a usinge-values/e-vectors. Then n = degpa (x) = am (x,)+- amcxid (cid:8869) v.
