In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The characteristic polynomial of an endomorphism of a finite-dimensional vector space is the characteristic polynomial of the matrix of that endomorphism over any base (that is, the characteristic polynomial does not depend on the choice of a basis). The characteristic equation, also known as the determinantal equation, is the equation obtained by equating the characteristic polynomial to zero.
In spectral graph theory, the characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix.
Motivation
In linear algebra, eigenvalues and eigenvectors play a fundamental role, since, given a linear transformation, an eigenvector is a vector whose direction is not changed by the transformation, and the corresponding eigenvalue is the measure of the resulting change of magnitude of the vector.
More precisely, if the transformation is represented by a square matrix an eigenvector and the corresponding eigenvalue must satisfy the equation
or, equivalently,
where is the identity matrix, and
(although the zero vector satisfies this equation for every it is not considered an eigenvector).
It follows that the matrix must be singular, and its determinant
must be zero.
In other words, the eigenvalues of are the roots of
which is a monic polynomial in of degree if is a matrix. This polynomial is the characteristic polynomial of .
Formal definition
Consider an matrix The characteristic polynomial of denoted by is the polynomial defined by
where denotes the identity matrix.
Some authors define the characteristic polynomial to be That polynomial differs from the one defined here by a sign so it makes no difference for properties like having as roots the eigenvalues of ; however the definition above always gives a monic polynomial, whereas the alternative definition is monic only when is even.
Examples
To compute the characteristic polynomial of the matrix
the determinant of the following is computed:
and found to be the characteristic polynomial of
Another example uses hyperbolic functions of a hyperbolic angle φ. For the matrix take
Its characteristic polynomial is
Properties
The characteristic polynomial of a matrix is monic (its leading coefficient is ) and its degree is The most important fact about the characteristic polynomial was already mentioned in the motivational paragraph: the eigenvalues of are precisely the roots of (this also holds for the minimal polynomial of but its degree may be less than ). All coefficients of the characteristic polynomial are polynomial expressions in the entries of the matrix. In particular its constant coefficient of is the coefficient of is one, and the coefficient of is , where is the trace of (The signs given here correspond to the formal definition given in the previous section; for the alternative definition these would instead be and respectively.)
For a matrix the characteristic polynomial is thus given by
Using the language of exterior algebra, the characteristic polynomial of an matrix may be expressed as
where is the trace of the th exterior power of which has dimension This trace may be computed as the sum of all principal minors of of size The recursive Faddeev–LeVerrier algorithm computes these coefficients more efficiently.
When the characteristic of the field of the coefficients is each such trace may alternatively be computed as a single determinant, that of the matrix,
The Cayley–Hamilton theorem states that replacing by in the characteristic polynomial (interpreting the resulting powers as matrix powers, and the constant term as times the identity matrix) yields the zero matrix. Informally speaking, every matrix satisfies its own characteristic equation. This statement is equivalent to saying that the minimal polynomial of divides the characteristic polynomial of
Two similar matrices have the same characteristic polynomial. The converse however is not true in general: two matrices with the same characteristic polynomial need not be similar.
The matrix and its transpose have the same characteristic polynomial. is similar to a triangular matrix if and only if its characteristic polynomial can be completely factored into linear factors over (the same is true with the minimal polynomial instead of the characteristic polynomial). In this case is similar to a matrix in Jordan normal form.