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Lecture

# Theorems

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University of Toronto St. George

Mathematics

MAT224H1

Martin, Burda

Winter

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Lecture 11 notes by Y. Burda We discussed the following theorems: Theorem. Let A be a real symmetric matrix. Then all its eigenvalues are real. Proof. If Av = v for some complex and v, then v Av = v v and also v Av = v A v = (Av) v = (v) v = v v. Thus = , i.e. is real. Theorem (Main theorem of symmetric matrices). A real matrix A is sym- 1 metric if and only if A = P DP for an orthogonal matrix P and diagonal matrix D, i.e. if and only if there exists an orthonormal basis of eigenvectors of A. Proof. If A = P 1DP for an orthogonal matrix P and diagonal matrix D, T T T 1T T 1 then A = P D P . Since P = P and D is symmetric (since it is diagonal), A = P 1 DP = A, i.e. A is symmetric. For the other direction we followed the book (theorem 8.19 on p.325) pretty closely. The main corollary of this theorem for us will be the following theorem: Theorem (Principal axis theorem). Let q be a real quadratic form on R . n n Then there exists an orthonormal basis of R so th

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