Eigenvector

Eigenvectors are a special set of vectors associated with a linear system of equations (i.e., a matrix equation) that are sometimes also known as characteristic vectors, proper vectors, or latent vectors (Marcus and Minc 1988, p. 144).

The determination of the eigenvectors and eigenvalues of a system is extremely important in physics and engineering, where it is equivalent to matrix diagonalization and arises in such common applications as stability analysis, the physics of rotating bodies, and small oscillations of vibrating systems, to name only a few. Each eigenvector is paired with a corresponding so-called eigenvalue. Mathematically, two different kinds of eigenvectors need to be distinguished: left eigenvectors and right eigenvectors. However, for many problems in physics and engineering, it is sufficient to consider only right eigenvectors. The term "eigenvector" used without qualification in such applications can therefore be understood to refer to a right eigenvector.

The decomposition of a square matrix into eigenvalues and eigenvectors is known in this work as eigen decomposition, and the fact that this decomposition is possible whenever the matrix consisting of the eigenvectors of is nonsingular is known as the eigen decomposition theorem.

Define a right eigenvector as a column vector satisfying

where is a matrix, so

which means the right eigenvalues must have zero determinant, i.e.,

Similarly, define a left eigenvector as a row vector satisfying

Taking the transpose of each side gives

which can be rewritten as

Rearrange again to obtain

which means

Rewriting gives

where the last step follows from the identity

Equations (◇) and (12) show that left and right eigenvalues are the same roots of the characteristic polynomial, a statement that is not true for eigenvectors.

Let be a matrix formed by the columns of the right eigenvectors and be a matrix formed by the rows of the left eigenvectors. Let

Then

and

so

If the eigenvalues in are distinct, this implies that is diagonal. More generally, it is block diagonal with respect to the degenerate eigenspaces, and bases within those eigenspaces can be chosen so that the left and right eigenvectors are biorthogonal. In particular, if is a symmetric matrix, then the left and right eigenvectors may be chosen to be simply each other's transpose, and if is a self-adjoint matrix (i.e., it is Hermitian), then they may be chosen to be adjoint matrices.

Eigenvectors may not be equal to the zero vector. A nonzero scalar multiple of an eigenvector is equivalent to the original eigenvector. Hence, without loss of generality, eigenvectors are often normalized to unit length.

While an matrix always has eigenvalues, some or all of which may be degenerate, such a matrix may have between 0 and linearly independent eigenvectors. For example, the matrix has only the single eigenvector .

Eigenvectors may be computed in the Wolfram Language using Eigenvectors[matrix]. This command always returns a list of length , so any eigenvectors that are not linearly independent are returned as zero vectors. Eigenvectors and eigenvalues can be returned together using the command Eigensystem[matrix].

Given a matrix with a vector basis of eigenvectors , , and and corresponding eigenvalues , , and , then an arbitrary vector can be written

Applying the matrix ,

so

If , and , it therefore follows that

so repeated application of the matrix to an arbitrary vector results, up to normalization, in a vector proportional to the eigenvector with eigenvalue of largest absolute value.