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.
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
It follows that the matrix must be singular, and its determinant
In other words, the eigenvalues of A are the roots of
Consider an matrix The characteristic polynomial of denoted by is the polynomial defined by
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
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 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 clearly is an eigenvalue of
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
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
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.
- Characteristic equation (disambiguation)
- Minimal polynomial (linear algebra)
- Invariants of tensors
- Companion matrix
- Faddeev–LeVerrier algorithm
- Cayley–Hamilton theorem
- Samuelson–Berkowitz algorithm
- Guillemin, Ernst (1953). Introductory Circuit Theory. Wiley. pp. 366, 541. ISBN 0471330663.
- 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.
- 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.
- "Characteristic Polynomial of a Graph – Wolfram MathWorld". Retrieved August 26, 2011.
- Steven Roman (1992). Advanced linear algebra (2 ed.). Springer. p. 137. ISBN 3540978372.
- Proposition 28 in these lecture notes[permanent dead link]
- Theorem 4 in these lecture notes
- 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.
- Lang, Serge (1993). Algebra. New York: Springer. p.567, Theorem 3.10. ISBN 978-1-4613-0041-0. OCLC 852792828.
- "secular equation". Retrieved January 21, 2010.
- 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 .
- Garibaldi, Skip (2004), "The characteristic polynomial and determinant are not ad hoc constructions", American Mathematical Monthly, 111 (9): 761–778, arXiv:math/0203276, doi:10.2307/4145188, JSTOR 4145188, MR 2104048
- 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 .