# spectral decomposition

(1.7 hours to learn)

## Summary

The spectral decomposition is the decomposition of a symmetric matrix A into QDQ^T, where Q is an orthogonal matrix and D is a diagonal matrix. The columns of Q correspond to the eigenvectors of A, and the diagonal entries of D correspond to the eigenvalues. This is possible because of a surprising fact about symmetric matrices: they have a full set of orthogonal eigenvectors. This decomposition gives a useful way to think about symmetric matrices: they are like diagonal matrices in a rotated coordinate system.

## Context

This concept has the prerequisites:

- matrix transpose (The spectral decomposition applies to symmetric matrices.)
- orthonormal bases (The spectral decomposition includes orthogonal factors.)
- change of basis (The spectral decomposition can be viewed as a change of basis.)
- matrix inverse (The spectral decomposition gives insight into the inverse of a symmetric matrix.)
- eigenvalues and eigenvectors (The spectral decomposition gives the eigenvectors and eigenvalues.)

## Core resources (read/watch one of the following)

## -Free-

→ MIT Open Courseware: Linear Algebra (2011)

Videos for an introductory linear algebra course focusing on numerical methods.

## -Paid-

→ Introduction to Linear Algebra

An introductory linear algebra textbook with an emphasis on numerical methods.

Location:
Section 6.4, "Symmetric matrices," pages 330-335

→ Multivariable Mathematics

A textbook on linear algebra and multivariable calculus with proofs.

Location:
Section 9.4, "The Spectral Theorem," pages 455-463

Additional dependencies:

- Lagrange multipliers

## See also

- The spectral decomposition is an example of diagonalization .
- The spectral decomposition can also be characterized as:
- the eigenvalues of a symmetric matrix
- the solution to an optimization problem

- A symmetric matrix is positive definite if its eigenvalues are positive