# 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.

The **characteristic polynomial of a graph** is the characteristic polynomial of its adjacency matrix. It is a graph invariant, though it is not complete: the smallest pair of non-isomorphic graphs with the same characteristic polynomial have five nodes.^{[1]}

## Contents

## 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 an eigenvector **v** ≠ 0 such that

or

(where * I* is the identity matrix). Since

**v**is non-zero, this means that the matrix

*λ*

*−*

**I***A*is singular (non-invertible), which in turn means that its determinant is 0. Thus the roots of the function det(

*λ*

*−*

**I***A*) are the eigenvalues of

*A*, and it is clear that this determinant is a polynomial in

*λ*.

## Formal definition

We consider an *n*×*n* matrix *A*. The characteristic polynomial of *A*, denoted by *p*_{A}(*t*), is the polynomial defined by

where *I* denotes the *n*-by-*n* identity matrix.

Some authors define the characteristic polynomial to be det(*A* - *t* *I*). 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 current definition always gives a monic polynomial, whereas the alternative definition always has constant term det(*A*).

## Examples

Suppose we want to compute the characteristic polynomial of the matrix

We 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}(*t*) 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 *p*_{A}(*t*) (this also holds for the minimal polynomial of *A*, but its degree may be less than *n*). The coefficients of the characteristic polynomial are all polynomial expressions in the entries of the matrix. In particular its constant coefficient *p*_{A} (0) is det(−*A*) = (−1)^{n} det(*A*), the coefficient of *t*^{n} is one, and the coefficient of *t*^{n−1} is tr(−*A*) = −tr(*A*), where tr(*A*) is the matrix trace of *A*. (The signs given here correspond to the formal definition given in the previous section;^{[2]} for the alternative definition these would instead be det(*A*) and (−1)^{n − 1} tr(*A*) respectively.^{[3]})

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(Λ^{k}A) 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 *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.

## 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 (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 *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*.

## 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, i.e. 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.
^{[4]}

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

## See also

- Characteristic equation (disambiguation)
- Minimal polynomial (linear algebra)
- Invariants of tensors
- Companion matrix
- Faddeev–LeVerrier algorithm
- Samuelson–Berkowitz algorithm

## References

- T.S. Blyth & E.F. Robertson (1998)
*Basic Linear Algebra*, p 149, Springer ISBN 3-540-76122-5 . - John B. Fraleigh & Raymond A. Beauregard (1990)
*Linear Algebra*2nd edition, p 246, Addison-Wesley ISBN 0-201-11949-8 . - Werner Greub (1974)
*Linear Algebra*4th edition, pp 120–5, Springer, ISBN 0-387-90110-8 . - Paul C. Shields (1980)
*Elementary Linear Algebra*3rd edition, p 274, Worth Publishers ISBN 0-87901-121-1 . -
Gilbert Strang (1988)
*Linear Algebra and Its Applications*3rd edition, p 246, Brooks/Cole ISBN 0-15-551005-3 .

## External links

- R. Skip Garibaldi. The characteristic polynomial and determinant are not ad hoc constructions. https://arxiv.org/abs/math/0203276