A linear algebra question?
Let A be a square matrix and let α be a scalar that is NOT an eigenvalue of A. Suppose that μ is an eigenvalue for the matrix B=(A-αI)^(-1) with corresponding eigenvector v. Prove that v is also an eigenvector for A and find a formula for the corresponding eigenvalue of A in terms of μ and α.
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μv = Bv = (A-αI)^(-1)v
(A-αI)Bv = v
ABv - αBv = v
A(μv) = αBv + v
μAv = (μα + I)v
Av = ((μα + I)/μ)v
Hence v is an eigenvector of A with eigenvalue ((μα + I)/μ)