Let A denote an n n matrix and put A1 A I, in R Show that is an eigenvalue of A if and only if is an eigenvalue of A1 How do the eigenvectors compare (Hence, the eigenvalues of A1 are just those of A shifted by )

Question: Let A denote an n n matrix and put A1 = A - I, in R. Show that is an eigenvalue of

Let A denote an n × n matrix and put A1 = A - αI, α in R. Show that λ is an eigenvalue of A if and only if λ - α is an eigenvalue of A1. How do the eigenvectors compare? (Hence, the eigenvalues of A1 are just those of A "shifted" by α.)

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