Suppose Ax x with x 0 Let a be a scalar different from the eigenvalues of A, and let B (A a1) 1 Sub tract ax from both sides of the equation Ax x, and use algebra to show that 1 ( a) is an eigenvalue of B, with x a corresponding eigenvector ...

Suppose If is not an eigenvalue of A then ...

Suppose Ax = x with x 0. Let a be a scalar different from the eigenvalues

Question:

Suppose Ax = λx with x ≠ 0. Let a be a scalar different from the eigenvalues of A, and let B = (A -a1)-1. Sub-tract ax from both sides of the equation Ax = λx, and use algebra to show that 1/ (λ - a) is an eigenvalue of B, with x a corresponding eigenvector.