Question: Prove Theorem 4.19. Suppose the n n matrix A has eigenvectors v 1 , v 2 , . . . , v m with

Prove Theorem 4.19.

Suppose the n × n matrix A has eigenvectors v1, v2, . . . , vm with corresponding eigenvalues λ1, λ2, . . . , λm. If x is a vector in Rn that can be expressed as a linear combination of these eigenvectors??say,

x = qv, + Gv2 + + GVm + GmVr

then, for any integer k,

image

x = qv, + Gv2 + + GVm + GmVr

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