Symmetric Matrices and Quadratic Forms

A real symmetric matrix always has an orthonormal basis of eigenvectors. This unit uses that fact to diagonalize symmetric matrices and to classify quadratic forms.

The spectral theorem is stated with its conditions, eigenvectors for distinct eigenvalues are proved orthogonal, and a repeated eigenvalue needs an orthonormal basis chosen inside its eigenspace. In coordinates along its principal axes, the directions of orthonormal eigenvectors of its matrix, a quadratic form has no cross terms, so the signs of the eigenvalues classify it as positive definite, negative definite, semidefinite or indefinite. A workshop then compares defective, rotation and symmetric examples and asks which method each one allows.