SYMMETRIC MATRICES AND QUADRATIC FORMS

Orthogonally Diagonalizable Symmetric Matrices and Eigenvalues How to Orthogonally Diagonalize Quadratic Forms Definiteness Principal Axes Ellipses and Hyperbolas

How to Orthogonally Diagonalize How to Examples Exercise How to: So, we want to orthogonally diagonalize a matrix A. The only stipulation we place on A is that it be symmetric. By Fact 2 we're guaranteed that A has N real eigenvalues. We can then regard the eigenvectors we find as a basis and use the Gram-Schmidt Process (or more simply QR-Factorialization) to convert this into an orthonormal basis. The matrix that results from this process will give us our orthogonally diagonalized matrix! Let's summarize the steps in MATLAB lingo:      Step 1: Make sure A is symmetric, that is A == A'      Step 2: Compute the eigenvectors of A, [V, D] = eig(A)      Step 3: Turn this into an orthonormal basis, [Q, R] = qr(V)      Step 4: Verify its correctness, show that inv(Q)*A*Q is a diagonal matrix.      Examples: Exercise: