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 vector spaces of finite dimension is the characteristic polynomial of the matrix of the endomorphism over any base; it does not depend on the choice of a basis. The characteristic equation, also known as the determinantal equation,[1][2][3] is the equation obtained by equating to zero the characteristic polynomial.

In spectral graph theory, the characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix.[4]

Given a square matrix A, we want to find a polynomial whose zeros are the eigenvalues of A. For a diagonal matrix A, the characteristic polynomial is easy to define: if the diagonal entries are a1a2a3, etc. then the characteristic polynomial will be:

This works because the diagonal entries are also the eigenvalues of this matrix.

For a general matrix A, one can proceed as follows. A scalar λ is an eigenvalue of A if and only if there is a nonzero vector v, called an eigenvector, such that

(where I is the identity matrix). Since v must be nonzero, this means that the matrix λIA has a nonzero kernel. Thus this matrix is not invertible, and the same is true for its determinant, which must therefore be zero. Thus the eigenvalues of A are the roots of det(λIA), which is a polynomial in λ.

We consider an n×n matrix A. The characteristic polynomial of A, denoted by pA(t), is the polynomial defined by[5]

Some authors define the characteristic polynomial to be det(AtI). That polynomial differs from the one defined here by a sign (−1)n, so it makes no difference for properties like having as roots the eigenvalues of A; however the definition above always gives a monic polynomial, whereas the alternative definition is monic only when n is even.

Suppose we want to compute the characteristic polynomial of the matrix

Another example uses hyperbolic functions of a hyperbolic angle φ. For the matrix take

The characteristic polynomial pA(t) of a n×n matrix is monic (its leading coefficient is 1) and its degree is n. The most important fact about the characteristic polynomial was already mentioned in the motivational paragraph: the eigenvalues of A are precisely the roots of pA(t) (this also holds for the minimal polynomial of A, but its degree may be less than n). All coefficients of the characteristic polynomial are polynomial expressions in the entries of the matrix. In particular its constant coefficient pA (0) is det(−A) = (−1)n det(A), the coefficient of tn is one, and the coefficient of tn−1 is tr(−A) = −tr(A), where tr(A) is the trace of A. (The signs given here correspond to the formal definition given in the previous section;[6] for the alternative definition these would instead be det(A) and (−1)n – 1 tr(A) respectively.[7])

Using the language of exterior algebra, one may compactly express the characteristic polynomial of an n×n matrix A as

When the characteristic of the field of the coefficients is 0, each such trace may alternatively be computed as a single determinant, that of the k×k matrix,

The Cayley–Hamilton theorem states that replacing t by A in the characteristic polynomial (interpreting the resulting powers as matrix powers, and the constant term c as c 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 A divides the characteristic polynomial of A.

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 A and its transpose have the same characteristic polynomial. A is similar to a triangular matrix if and only if its characteristic polynomial can be completely factored into linear factors over K (the same is true with the minimal polynomial instead of the characteristic polynomial). In this case A is similar to a matrix in Jordan normal form.

If A and B are two square n×n matrices then characteristic polynomials of AB and BA coincide:

When A is non-singular this result follows from the fact that AB and BA are similar:

For the case where both A and B are singular, one may remark that the desired identity is an equality between polynomials in t 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 A is a matrix of order m×n and B is a matrix of order n×m, then AB is m×m and BA is n×n matrix, and one has

To prove this, one may suppose n > m, by exchanging, if needed, A and B. Then, by bordering A on the bottom by nm rows of zeros, and B on the right, by, nm columns of zeros, one gets two n×n matrices A' and B' such that B'A' = BA, and A'B' is equal to AB bordered by nm rows and columns of zeros. The result follows from the case of square matrices, by comparing the characteristic polynomials of A'B' and AB.

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, i.e. slow compared to annual motion) of planetary orbits, according to Lagrange's theory of oscillations.