Lecture Notes

11. Eigenvectors and Diagonalization

Part of the Series on Linear Algebra.

By Akshay Agrawal. Last updated Dec. 20, 2018.

Previous entry: Isometries ; Next entry: Symmetric Matrices

In this section, we introduce the concepts of eigenvectors and eigenvalues of square matrices. The eigenvectors and eigenvalues of a matrix provide insight into its behavior as an operator; for this reason, the list of eigenvalues of a matrix is referred to as its spectrum. An important result that we will present is that if a matrix has a basis of eigenvectors, then it is diagonal with respect to that basis.

Definition

A nonzero vector is an eigenvector of a matrix if there exists a scalar such that

The scalar is referred to as the eigenvalue corresponding to the eigenvector . Notice that is an eigenvalue of if and only if is not injective, or, equivalently, is not invertible.

It is a fact that eigenvectors corresponding to distinct eigenvalues are linearly independent; for a proof, see 5.10 of Axler. Notice that this means that has at most distinct eigenvalues.

Diagonalizable matrices

Let be a basis of consisting of eigenvectors of , with corresponding eigenvalues (note that such a basis need not exist). Then has a diagonal matrix with respect to the basis :

Whenever has a diagonal matrix with respect to some basis, we say that is diagonalizable. Concretely, letting , we have that

or equivalently that

The matrix is referred to as an eigendecomposition of .

Diagonalizability of is equivalent to having linearly independent eigenvectors. We can also give a sufficient (but not necessary) condition for diagonalizability: any matrix with distinct eigenvalues is diagonalizable, since eigenvectors corresponding to distinct eigenvalues are linearly independent.

Notice that it is easy to compute powers of diagonalizable matrices:

Relationship between eigenvalues and rank

The rank of an matrix is equal to the number of nonzero eigenvalues; in other words, if the eigenvalue is repeated times, the rank of the matrix is . The eigendecomposition of a diagonalizable matrix of rank is of the form

References

  1. Sheldon Axler. Linear Algebra Done Right.
  2. Stephen Boyd and Sanjay Lall. EE 263 Course Notes.