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04-28-03 - Lecture 31 Andrei Antonenko 1 Symmetric matrices...

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Lecture 31 Andrei Antonenko April 28, 2003 1 Symmetric matrices As we saw before, sometimes it happens that the matrix is not diagonalizable over R .. It may happen, for example, when some roots of the characteristic polynomial are complex. In this lecture we will consider the properties of symmetric matrices. Theorem 1.1. Let A be a real symmetric matrix. Then each root λ of its characteristic polynomial is real. Moreover, for symmetric matrices, eigenvectors, corresponding to different eigenvalues are not only linearly independent, but even orthogonal: Theorem 1.2. Let A be a real symmetric matrix. Let λ 1 and λ 2 be different eigenvalues of A , and e 1 and e 2 are corresponding eigenvectors. Then e 1 and e 2 are orthogonal, i.e. h e 1 , e 2 i = 0 . So, these 2 theorems give us the following main result about symmetric matrices: Theorem 1.3. Let A be a real symmetric matrix. Then there exists a matrix C such that D = C - 1 AC is diagonal. Moreover, columns of a matrix C are orthogonal.
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