# Eigenvectors and eigenvalues

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Suppose that A and B are n x n matrices such that A = SBS-1 (where S is an invertible matrix), so A and B are similar matrices.

(a) Show: if v is an eigenvector of B with eigenvalue μ, then Sv is an eigenvector for A.

(Remember that only nonzero vectors can be eigenvectors.)

What is the corresponding eigenvalue?

(b) Suppose that w is an eigenvector of A with eigenvalue β. Find an eigenvector for B which has the same eigenvalue β.

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This provides examples of working with eigenvectors and eigenvalues of matrices.

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Proof:

(a) If is an eigenvector of with eigenvalue , ...

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