Polynomial whose roots are the eigenvalues of a matrix
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.
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.
To compute the characteristic polynomial of the matrix
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 is the coefficient of is one, and the coefficient of is tr(−A) = −tr(A), where tr(A) 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 (−1)n – 1 tr(A) 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
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 matrixif 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.
Characteristic polynomial of a product of two matrices
If and are two square matrices then characteristic polynomials of and coincide:
For the case where both and are singular, the desired identity is an equality between polynomials in and the coefficients of the matrices. Thus, to prove this equality, it suffices to prove that it is verified on a non-empty open subset (for the usual topology, or, more generally, for the Zariski topology) of the space of all the coefficients. As the non-singular matrices form such an open subset of the space of all matrices, this proves the result.
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 Ak
If is an eigenvalue of a square matrix with eigenvector then is an eigenvalue of because
The multiplicities can be shown to agree as well, and this generalizes to any polynomial in place of :
Theorem — Let be a square matrix and let be a polynomial. If the characteristic polynomial of has a factorization
then the characteristic polynomial of the matrix is given by
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.
This proof only applies to matrices and polynomials over complex numbers (or any algebraically closed field).
In that case, the characteristic polynomial of any square matrix can be always factorized as
where are the eigenvalues of possibly repeated.
Moreover, the Jordan decomposition theorem guarantees that any square matrix can be decomposed as where is an invertible matrix and is upper triangular
with on the diagonal (with each eigenvalue repeated according to its algebraic multiplicity).
(The Jordan normal form has stronger properties, but these are sufficient; alternatively the Schur decomposition can be used, which is less popular but somewhat easier to prove).
For an upper triangular matrix with diagonal the matrix is upper triangular with diagonal in
and hence is upper triangular with diagonal
Therefore, the eigenvalues of are
Since is similar to it has the same eigenvalues, with the same algebraic multiplicities.
Secular function and secular equation
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 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
The above definition of the characteristic polynomial of a matrix with entries in a field generalizes without any changes to the case when is just a commutative ring. Garibaldi (2004) defines the characteristic polynomial for elements of an arbitrary finite-dimensional (associative, but not necessarily commutative) algebra over a field and proves the standard properties of the characteristic polynomial in this generality.