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 basis (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

Eigenvalues and eigenvectors play a fundamental role in linear algebra, 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.

is the identity matrix it is not considered an eigenvector).

must be singular must be zero.

In other words, the eigenvalues of are the roots which is a monic polynomial in of degree if is a matrix. This polynomial is the characteristic polynomial of .

Formal definition

is the polynomial defined by identity matrix.

; however the definition above always gives a monic polynomial is even.

Examples

the determinant

Another example uses hyperbolic functions of a hyperbolic angle

Properties

is monic The most important fact about the characteristic polynomial was already mentioned in the motivational paragraph are precisely the roots (this also holds for the minimal polynomial). All coefficients of the characteristic polynomial are polynomial expressions is , where is the trace (The signs given here correspond to the formal definition and respectively.)

Using the language of exterior algebra is the traceth exterior power This trace may be computed as the sum of all principal minors The recursive Faddeev–LeVerrier algorithm computes these coefficients more efficiently .

When the characteristic of the field

The Cayley–Hamilton theorem 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

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.

and its transpose is similar to a triangular matrix if and only if is similar to a matrix in Jordan normal form.

Characteristic polynomial of a product of two matrices

Characteristic polynomial of Ak

:

The theorem applies to matrices and polynomials over any field or commutative ring. has a factorization into linear factors is not always true, unless the matrix is over an algebraically closed field such as the complex numbers.

Secular function and secular equation

Secular function

The term secular function has been used for what is now called characteristic polynomial (in some literature the term secular function is still used). The term comes from the fact that the characteristic polynomial was used to calculate secular perturbations (on a time scale of a century, that is, slow compared to annual motion) of planetary orbits, according to Lagrange's theory of oscillations.

Secular equation

Secular equation may have several meanings.

• In linear algebra it is sometimes used in place of characteristic equation.

• In astronomy it is the algebraic or numerical expression of the magnitude of the inequalities in a planet's motion that remain after the inequalities of a short period have been allowed for.

• In molecular orbital calculations relating to the energy of the electron and its wave function it is also used instead of the characteristic equation.

For general associative algebras

is just a commutative ring. defines the characteristic polynomial for elements of an arbitrary finite-dimensional (associative and proves the standard properties of the characteristic polynomial in this generality.

Theoretical complexity: calculation by fast matrix multiplication

It is possible to calculate the characteristic polynomial in a fast way with the use of fast matrix multiplication algorithms slightly above 2.37. Respective algorithms is given by.

See also

• Characteristic equation (disambiguation)

• Invariants of tensors

• Companion matrix

• Faddeev–LeVerrier algorithm

• Cayley–Hamilton theorem

• Samuelson–Berkowitz algorithm

References

• T.S. Blyth & E.F. Robertson (1998) Basic Linear Algebra, p 149, Springer .

• John B. Fraleigh & Raymond A. Beauregard (1990) Linear Algebra 2nd edition, p 246, Addison-Wesley .

• Werner Greub (1974) Linear Algebra 4th edition, pp 1205, Springer, .

• Paul C. Shields (1980) Elementary Linear Algebra 3rd edition, p 274, Worth Publishers .

• Gilbert Strang (1988) Linear Algebra and Its Applications 3rd edition, p 246, Brooks/Cole .