Characteristic polynomial
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. 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
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.More precisely, suppose the transformation is represented by a square matrix Then an eigenvector and the corresponding eigenvalue must satisfy the equation
or, equivalently,
where is the identity matrix, and
.
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 bywhere 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 matrixthe 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 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 . 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 1, and the coefficient of is, where is the trace ofFor 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 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 . In this case is similar to a matrix in Jordan normal form.
Characteristic polynomial of a product of two matrices
If and are two square matrices then characteristic polynomials of and coincide:Proof: If is a non-zero generalized eigenvalue of of algebraic multiplicity, and belongs to the kernel of, then belongs to the kernel of, so the non-zero generalized eigenspaces of and have the same dimension. Therefore, since and are both, the remaining generalized eigenspaces, with eigenvalue 0, have the same dimension. Therefore and have the same characteristic polynomial, because all generalized eigenvalues are the same, with the same algebraic multiplicities.
More generally, if is a matrix of order and is a matrix of order then is and is matrix, and one has
To prove this, one may suppose by exchanging, if needed, and Then, by bordering on the bottom by rows of zeros, and on the right, by, columns of zeros, one gets two matrices and such that and is equal to bordered by rows and columns of zeros. The result follows from the case of square matrices, by comparing the characteristic polynomials of and
Characteristic polynomial of ''A''''k''
If is an eigenvalue of a square matrix with eigenvector then is an eigenvalue of becauseThe multiplicities can be shown to agree as well, and this generalizes to any polynomial in place of :
That is, the algebraic multiplicity of in equals the sum of algebraic multiplicities of in over such that
In particular, and
Here a polynomial for example, is evaluated on a matrix simply as
The theorem applies to matrices and polynomials over any field or commutative ring.
However, the assumption that 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. The term comes from the fact that the characteristic polynomial was used to calculate secular perturbations 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