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Unformatted text preview: AIMS Lecture Notes 2006 Peter J. Olver 6. Eigenvalues and Singular Values In this section, we collect together the basic facts about eigenvalues and eigenvectors. From a geometrical viewpoint, the eigenvectors indicate the directions of pure stretch and the eigenvalues the extent of stretching. Most matrices are complete, meaning that their (complex) eigenvectors form a basis of the underlying vector space. A particularly important class are the symmetric matrices, whose eigenvectors form an orthogonal basis of R n . A non-square matrix A does not have eigenvalues. In their place, one uses the square roots of the eigenvalues of the associated square Gram matrix K = A T A , which are called singular values of the original matrix. The numerical computation of eigenvalues and eigenvectors is a challenging issue, and must be be deferred until later. 6.1. Eigenvalues and Eigenvectors. We inaugurate our discussion of eigenvalues and eigenvectors with the basic definition. Definition 6.1. Let A be an n n matrix. A scalar is called an eigenvalue of A if there is a non-zero vector v 6 = , called an eigenvector , such that A v = v . (6 . 1) In other words, the matrix A stretches the eigenvector v by an amount specified by the eigenvalue . Remark : The odd-looking terms eigenvalue and eigenvector are hybrid German English words. In the original German, they are Eigenwert and Eigenvektor , which can be fully translated as proper value and proper vector. For some reason, the half- translated terms have acquired a certain charm, and are now standard. The alternative English terms characteristic value and characteristic vector can be found in some (mostly older) texts. Oddly, the term characteristic equation , to be defined below, is still used. The requirement that the eigenvector v be nonzero is important, since v = is a trivial solution to the eigenvalue equation (6.1) for any scalar . Moreover, as far as solving linear ordinary differential equations goes, the zero vector v = gives u ( t ) , which is certainly a solution, but one that we already knew. The eigenvalue equation (6.1) is a system of linear equations for the entries of the eigenvector v provided that the eigenvalue is specified in advance but is mildly 3/15/06 86 c 2006 Peter J. Olver nonlinear as a combined system for and v . Gaussian Elimination per se will not solve the problem, and we are in need of a new idea. Let us begin by rewriting the equation in the form ( A- I ) v = , (6 . 2) where I is the identity matrix of the correct size . Now, for given , equation (6.2) is a homogeneous linear system for v , and always has the trivial zero solution v = . But we are specifically seeking a nonzero solution! A homogeneous linear system has a nonzero solution v 6 = if and only if its coefficient matrix, which in this case is A- I , is singular....
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