Linear Algebra · Chapter 11

Symmetric matrices and the spectral theorem

Where eigenvalues and orthogonality meet: symmetric matrices, their orthonormal bases of eigenvectors, the quadratic forms they describe, and a review of the whole course.

A = Q Λ Q^{T}

A = A^{T}

Q^{T} Q = I

x^{T} A x

λ_{i} \in R

Λ = diag (λ_{1}, \dots, λ_{n})

q_{i}^{T} q_{j} = δ_{ij}

x = Q y

Lessons

Build the foundation

01Symmetric matricesA matrix equal to its own transpose is the best behaved of all square matrices. Its eigenvalues are real, and eigenvectors for different eigenvalues are at right angles, so it stretches space along perpendicular directions and never turns or shears it.Read lesson →
02The spectral theoremEvery symmetric matrix can be diagonalised, and better than that, by a rotation or reflection of the axes: it has an orthonormal basis of eigenvectors. In that basis it does nothing but stretch each axis by its eigenvalue. Conversely, only symmetric matrices can be described that way.Read lesson →
03Quadratic formsAn expression in which every term has degree two, such as three x squared plus two x y plus three y squared, is a quadratic form. The cross term makes it hard to read, but turning the axes onto the eigenvectors of its symmetric matrix removes the cross term, and what is left shows at once its sign, its largest and smallest values and the shape of its level curves.Read lesson →
04Course reviewThe whole course in one sitting: systems and matrices, determinants and inverses, elimination, vector spaces and linear maps, eigenvalues, orthogonality and least squares, and the spectral theorem. Each question needs a different chapter, and no worked example sits above the answers.Read lesson →

Mapping the lesson.

Symmetric matrices and the spectral theorem

Build the foundation