Unit 6 of 10

Quadratic Forms

  • 4 lessons
  • Videos 27 to 30

This unit classifies quadratic forms by the signs of their values and describes the eigenvalues of a Hermitian matrix through minimum and maximum problems.

Sylvester's law of inertia shows that the numbers of positive, negative and zero coefficients in a diagonal form do not depend on the congruence used to reach it. Every positive definite matrix has a unique Cholesky factorization whose triangular factor has a positive diagonal, and a Hermitian matrix is positive definite exactly when all its leading principal minors are positive. Rayleigh quotients give the smallest and largest eigenvalues, and the min-max principle describes every eigenvalue in between.

Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Inertia and Positive Definite Factorization

  1. Quadratic Forms and Sylvester's Law of Inertia

    How a real symmetric or complex Hermitian form reduces by congruence to a diagonal of ones, minus ones and zeros, why these counts, the inertia, never change, and what makes a form indefinite.

    Video 27

  2. Positive Definite Matrices and Cholesky Factorization

    What positive definite and semidefinite matrices are, how completing the square and the Schur complement build the Cholesky factorization, and why leading principal minors test a Hermitian matrix for positive definiteness.

    Video 28

Rayleigh Quotients and Min-Max Bounds

  1. Rayleigh Quotients

    How the Rayleigh quotient of a Hermitian matrix ignores scaling, equals a weighted average of eigenvalues, and has the smallest and largest eigenvalues as its minimum and maximum, attained on whole eigenspaces.

    Video 29

  2. The Min-Max Principle

    How the min-max and max-min principles describe every eigenvalue of a Hermitian matrix in increasing order, proved by a dimension count for intersecting subspaces and an attaining subspace.

    Video 30