MATB24H3 Lecture Notes - Orthogonal Matrix, Symmetric Matrix, Row And Column Vectors
Document Summary
Theorem: properties of ax for an orthogonal matrix alet a be an orthogonal n n matrix and x and y are column vectors in rn. Then: (ax) (ay) = x y preservation of dot product, ||ax|| = ||x|| preservation of length, the angle between nonzero vectors x and y equals the angle between. Showing that is also the angle between the nonzero vectors x and y. Theorem: orthogonality of eigenspaces of a real symmetric matrix: Let a be a real symmetric matrix and 1 and 2 be distinct eigenvalues of: then the eigenspaces e 1 and e 2 are orthogonal. proof. : to show that e 1 and e 2 are orthogonal we must prove v1 and v2 are orthogonal for all v1 e 1 and v2 e 2. Let v1 e 1 and v2 e 2. 1(v1 v2) = 2(v1 v2) or ( 1 2)(v1 v2) = 0.
