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 base (that is, the characteristic polynomial 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 the characteristic polynomial to zero.

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

MotivationEdit

In linear algebra, eigenvalues and eigenvectors play a fundamental role, 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, if the transformation is represented by a square matrix   an eigenvector   and the corresponding eigenvalue   must satisfy the equation

 
or, equivalently,
 
where   is the identity matrix, and   (although the zero vector satisfies this equation for every   it is not considered as an eigenvector).

It follows that the matrix   must be singular, and its determinant

 
must be zero.

In other words, the eigenvalues of A are the roots of

 
which is a monic polynomial in x of degree n if A is a n×n matrix. This polynomial is the characteristic polynomial of A.

Formal definitionEdit

Consider an   matrix   The characteristic polynomial of   denoted by   is the polynomial defined by[5]

 
where   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.

ExamplesEdit

To compute the characteristic polynomial of the matrix

 
the 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
 

PropertiesEdit

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;[6] for the alternative definition these would instead be   and (−1)n – 1 tr(A) respectively.[7])

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

 
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 (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 matrix if 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 matricesEdit

If   and   are two square   matrices then characteristic polynomials of   and   coincide:

 

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

 

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 AkEdit

If   is an eigenvalue of a square matrix   with eigenvector   then clearly   is an eigenvalue of  

 

The multiplicities can be shown to agree as well, and this generalizes to any polynomial in place of  :[8]

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.[9] 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.

Proof

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).

Let   Then

 
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 equationEdit

Secular functionEdit

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 equationEdit

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.[10]
  • 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 algebrasEdit

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.

See alsoEdit

ReferencesEdit

  1. ^ Guillemin, Ernst (1953). Introductory Circuit Theory. Wiley. pp. 366, 541. ISBN 0471330663.
  2. ^ Forsythe, George E.; Motzkin, Theodore (January 1952). "An Extension of Gauss' Transformation for Improving the Condition of Systems of Linear Equations" (PDF). American Mathematical Society – Mathematics of Computation. 6 (37): 18–34. doi:10.1090/S0025-5718-1952-0048162-0. Retrieved 3 October 2020.
  3. ^ Frank, Evelyn (1946). "On the zeros of polynomials with complex coefficients". Bulletin of the American Mathematical Society. 52 (2): 144–157. doi:10.1090/S0002-9904-1946-08526-2.
  4. ^ "Characteristic Polynomial of a Graph – Wolfram MathWorld". Retrieved August 26, 2011.
  5. ^ Steven Roman (1992). Advanced linear algebra (2 ed.). Springer. p. 137. ISBN 3540978372.
  6. ^ Proposition 28 in these lecture notes[permanent dead link]
  7. ^ Theorem 4 in these lecture notes
  8. ^ Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis (2nd ed.). Cambridge University Press. pp. 108–109, Section 2.4.2. ISBN 978-0-521-54823-6.
  9. ^ Lang, Serge (1993). Algebra. New York: Springer. p.567, Theorem 3.10. ISBN 978-1-4613-0041-0. OCLC 852792828.
  10. ^ "secular equation". Retrieved January 21, 2010.