How Can I be able to verify the eigenvectors of a given 3x3 matrix iare Orthogonal? And In more general sense supposing this 3x3 matrics is Given A=[-2 1 0; -4 0 3; 0 1 -2] then I wanted to figure out an orthogonal matrices P such that P^t A P is upper triangular matrices. t = transpose! Tanx In Advance

Question

How Can I be able to verify the eigenvectors of a given 3x3 matrix iare Orthogonal? And In more general sense supposing this 3x3 matrics  is Given A=[-2 1 0; -4 0 3; 0 1 -2] then I wanted to figure out an orthogonal matrices P such that P^t A P is upper triangular matrices. t = transpose! Tanx In Advance

anonymous · Answer

If the inner (dot) product of two vectors evaluates to zero, they're orthogonal. This must be true for each pair of vectors for the matrix to be orthogonal. If you have a real symmetric matrix, the eigenvalues are real and the eigenvectors will be orthogonal (you can normalize to get orthonormal eigenvectors). This is the famous diagonalization of a matrix A=QDQ^T, where Q is orthonormal and D is a diagonal matrix whose diagonal entries are the eigenvalues. Another useful factorization is A=QR, known as the Grant-Schmidt factorization of a matrix. Q is orthonormal, R is upper triangular (refer to chapter 4 of Strang's Intro to Linear Algebra text book)