eigenvectors and eigenvalues
eigenvectors and eigenvalues
eigenvectors and eigenvalues
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Solution:2 0 0___ 0 0A2I 1 3 10 ___ 01 1 30 0 ___2 ___ 0 01 3 ___ 11 1 3 ______ 0 01 ___ 11 1 ___Augmented matrix for A2Ix 0:0 0 0 01 1 1 01 1 1 01 1 1 00 0 0 00 0 0 0x 1x 2 x 311x x 2x 2 ____1 ____0x 3x 3015
EXAMPLE: Suppose is eigenvalue of A. Determine aneigenvalue of A 2 <strong>and</strong> A 3 . In general, what is an eigenvalue of A n ?Solution: Since is eigenvalue of A, there is a nonzero vector xsuch thatAx x.Then___Ax ___xA 2 x AxA 2 x ___xA 2 x 2 xTherefore 2 is an eigenvalue of A 2 .Show that 3 is an eigenvalue of A 3 :___A 2 x ___ 2 xA 3 x 2 AxA 3 x 3 xTherefore 3 is an eigenvalue of A 3 .In general, ______ is an eigenvalue of A n .7
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