# diagonalization

(1.3 hours to learn)

## Summary

Diagonalization refers to factorizing a matrix as A = SDS^-1, where D is a diagonal matrix. The entries of D correspond to the eigenvalues of A, and the columns of S correspond to the eigenvectors. The diagonal representation is useful for computing powers of matrices. Unfortunately, not all matrices are diagonalizable.

## Context

This concept has the prerequisites:

- matrix multiplication (Diagonalization is a kind of matrix factorization.)
- matrix inverse (The diagonalized representation includes an inverse.)
- eigenvalues and eigenvectors (The matrices in the factorization represent the eigenvalues and eigenvectors of the matrix.)
- bases (Diagonalization gives a basis for the space in terms of the eigenvectors.)

## 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.

→ A First Course in Linear Algebra (2012)

A linear algebra textbook with proofs.

Location:
Section "Similarity and diagonalization"

Additional dependencies:

- complex vectors and matrices

## -Paid-

→ Multivariable Mathematics

A textbook on linear algebra and multivariable calculus with proofs.

Location:
Section 9.2, "Eigenvalues, eigenvectors, and applications," subsection 9.2.2, "Diagonalizability"

Additional dependencies:

- linear transformations as matrices

→ Introduction to Linear Algebra

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

Location:
Section 6.2, "Diagonalizing a matrix"

## See also

-No Additional Notes-