SYMMETRIC MATRICES AND QUADRATIC FORMS

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

Symmetric Matrices and Eigenvalues How to Examples Exercise How to: There are a couple of interesting facts about symmetric matrices and eigenvalues that are useful in orthogonal diagonalization. Fact 1: If A is a symmetric matrix and v1 and v2 are eigenvectors whose eigenvalues are distinct (distinct means they are both different), then dot(v1, v2) = 0 or in other words they are orthogonal vectors. Fact 2: An NxN symmetric matrix has N real eigenvalues if they are counted with their algebraic multiplicities. Algebraic multiplicty is kind of like repeated roots. If the term (lambda - 1)^3 appears in our characteristic polynomial, then we say the value 1 is a three times repeated root, or in the case of eigenvalues, lambda = 1 has an algebraic multiplicity of 3. Now we're ready to orthogonally diagonalize symmetric matrices.      Examples: Exercise: