# 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 as 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**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.

## Motivation

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

*a*

_{1},

*a*

_{2},

*a*

_{3}, 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

or, equivalently,

. Since must be non-zero, this means that the matrix 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 are the roots of, which is a polynomial in.

## Formal definition

We consider an*n*×

*n*matrix

*A*. The characteristic polynomial of

*A*, denoted by

*p*, is the polynomial defined by

_{A}where

*I*denotes the

*n*×

*n*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

*A*; however the definition above always gives a monic polynomial, whereas the alternative definition is monic only when

*n*is even.

## Examples

Suppose we want to compute the characteristic polynomial of the matrixWe now compute the determinant of

which is the characteristic polynomial of

*A*.

Another example uses hyperbolic functions of a hyperbolic angle φ.

For the matrix take

Its characteristic polynomial is

## Properties

The polynomial*p*

_{A}is monic 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

*p*

_{A}. The coefficients of the characteristic polynomial are all polynomial expressions in the entries of the matrix. In particular its constant coefficient

*p*

_{A}is det =

^{n}det, the coefficient of is one, and the coefficient of is tr = −tr, where is the trace of

*A*. and

^{n − 1 }tr

For a 2×2 matrix

*A*, the characteristic polynomial is thus given by

Using the language of exterior algebra, one may compactly express the characteristic polynomial of an

*n*×

*n*matrix

*A*as

where tr is the trace of the

*k*

^{th}exterior power of

*A*, which has dimension. This trace may be computed as the sum of all principal minors of

*A*of size

*k*. The recursive Faddeev–LeVerrier algorithm computes these coefficients more efficiently.

When the characteristic is 0 it may alternatively be computed as a single determinant, that of the matrix,

The Cayley–Hamilton theorem states that replacing

*t*by

*A*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

*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*. In this case

*A*is similar to a matrix in Jordan normal form.

## Characteristic polynomial of a product of two matrices

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 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

*n*–

*m*rows of zeros, and

*B*on the right, by,

*n*–

*m*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

*n*–

*m*rows and columns of zeros. The result follows from the case of square matrices, by comparing the characteristic polynomials of

*A'B'*and

*AB*.

## Characteristic polynomial of ''A''^{''k''}

If is an eigenvalue of a square matrix *A*with eigenvector

**v**, then clearly is an eigenvalue of

*A*

^{k}

The 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

*A*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.

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.

.

Let.

Then

It is easy to check 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 mulitiplicities.

## 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

*F*generalizes without any changes to the case when

*F*is just a commutative ring. defines the characteristic polynomial for elements of an arbitrary finite-dimensional algebra over a field

*F*and proves the standard properties of the characteristic polynomial in this generality.