MAT224H1 Lecture : Theorems
Document Summary
If av = v for some complex and v, then v av = v v and also v av = v a v = (av) v = ( v)t v = v v. Thus = , i. e. is real. A real matrix a is sym- metric if and only if a = p 1dp for an orthogonal matrix p and diagonal matrix d, i. e. if and only if there exists an orthonormal basis of eigenvectors of a. If a = p 1dp for an orthogonal matrix p and diagonal matrix d, then at = p t dt p 1t. Since p t = p 1 and d is symmetric (since it is diagonal), at = p 1dp = 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: Let q be a real quadratic form on rn.
