Question

Let ( lambda ) be an eigenvalue of the ( n times n ) matrix ( A ). Define ( A^{2} ) to be the ( n times n ) matrix ( A^{2}=A A ). Show that ( lambda^{2} ) is an eigenvalue of ( A^{2} ).

Answer 1

To prove lambda squared is an eigenvalue of A squared, one must find a vector v for which Av = lambda v and then apply A again to both sides, yielding A squared v = lambda squared v.

To show that if lambda is an eigenvalue of the matrix A, then lambda2 is an eigenvalue of A2 (where A2 = A A), we must find a vector v such that Av = lambdav. If we then apply A to both sides of the equation, we get A2v = A(lambdav) = lambda(Av) = lambda2v. This shows that v is also an eigenvector of A2 with the eigenvalue lambda2.