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Unformatted text preview: Statistics 351 (Fall 2007) Review of Linear Algebra Suppose that A is the symmetric matrix A = 1- 1 0- 1 2 1 1 3 . Determine the eigenvalues and eigenvectors of A . Recall that a real number λ is an eigenvalue of A if A v = λ v for some vector v 6 = 0. We call v an eigenvector (corresponding to the eigenvalue λ ) of A . Note that if v is an eigenvector of A , then so too is α v for any non-zero real number α . The non-zero vector v is a solution of the equation A v = λ v if and only if v is also a solution of the equation ( A- λI ) v = 0. The equation ( A- λI ) v = 0 has a non-zero solution if and only if the matrix A- λI is singular (non-invertible). The matrix A- λI is invertible if and only if det[ A- λI ] 6 = 0. Therefore, in order to find the eigenvalues of A , we need to find those values of λ such that det[ A- λI ] = 0. (We sometimes call the polynomial equation det[ A- λI ] = 0 the characteristic equation of the matrix A .) Therefore, we consider A- λI = 1- λ- 1- 1 2- λ 1 1 3- λ ....
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