In
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices ...
, the characteristic polynomial of a
square matrix is a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
which is invariant under
matrix similarity
In linear algebra, two ''n''-by-''n'' matrices and are called similar if there exists an invertible ''n''-by-''n'' matrix such that
B = P^ A P .
Similar matrices represent the same linear map under two (possibly) different bases, with being ...
and has the
eigenvalues as
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
. It has the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
and the
trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''Trace'' ...
of the matrix among its coefficients. The characteristic polynomial of an
endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a gr ...
of a finite-dimensional
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
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
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
). The characteristic equation, also known as the determinantal equation, is the equation obtained by equating the characteristic polynomial to zero.
In
spectral graph theory, the characteristic polynomial of a
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
is the characteristic polynomial of its
adjacency matrix
In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph.
In the special case of a finite simp ...
.
Motivation
In
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices ...
,
eigenvalues and eigenvectors
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
play a fundamental role, since, given a
linear transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
, 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 are the
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
of
which is a
monic polynomial
In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form:
:x^n+c_x^+\c ...
in of degree if is a matrix. This polynomial is the ''characteristic polynomial'' of .
Formal definition
Consider an
matrix
The characteristic polynomial of
denoted by
is the polynomial defined by
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
In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form:
:x^n+c_x^+\c ...
, whereas the alternative definition is monic only when
is even.
Examples
To compute the characteristic polynomial of the matrix
the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of the following is computed:
and found to be
the characteristic polynomial of
Another example uses
hyperbolic function
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...
s of a
hyperbolic angle
In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of ''xy'' = 1 in Quadrant I of the Cartesian plane. The hyperbolic angle parametrises the unit hyperbola, which has hyperbolic function ...
φ.
For the matrix take
Its characteristic polynomial is
Properties
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
root
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the su ...
s 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 , where is the
trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''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 respectively.)
For a
matrix
the characteristic polynomial is thus given by
Using the language of
exterior algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
, the characteristic polynomial of an
matrix
may be expressed as
where
is the
trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''Trace'' ...
of the
th
exterior power
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
of
which has dimension
This trace may be computed as the sum of all
principal minor
In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix, cut down from A by removing one or more of its rows and columns. Minors obtained by removing just one row and one column from square matrices (first minors ...
s of
of size
The recursive
Faddeev–LeVerrier algorithm
In mathematics (linear algebra), the Faddeev–LeVerrier algorithm is a recursive method to calculate the coefficients of the characteristic polynomial p_A(\lambda)=\det (\lambda I_n - A) of a square matrix, , named after Dmitry Konstantinovi ...
computes these coefficients more efficiently.
When the
characteristic of the
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
of the coefficients is
each such trace may alternatively be computed as a single determinant, that of the
matrix,
The
Cayley–Hamilton theorem
In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) satisfies ...
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
In linear algebra, two ''n''-by-''n'' matrices and are called similar if there exists an invertible ''n''-by-''n'' matrix such that
B = P^ A P .
Similar matrices represent the same linear map under two (possibly) different bases, with being ...
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
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
have the same characteristic polynomial.
is similar to a
triangular matrix
In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal are ...
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
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
In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF),
is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to so ...
.
Characteristic polynomial of a product of two matrices
If
and
are two square
matrices then characteristic polynomials of
and
coincide:
When
is
non-singular
In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In ca ...
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
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, or, more generally, for the
Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
) 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 ''A''''k''
If
is an eigenvalue of a square matrix
with eigenvector
then
is an eigenvalue of
because
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
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.
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, that is, 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
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices ...
it is sometimes used in place of characteristic equation.
* In
astronomy
Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
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
In chemistry, a molecular orbital is a mathematical function describing the location and wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of findin ...
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. 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 also
*
Characteristic equation (disambiguation)
*
monic polynomial (linear algebra)
*
Invariants of tensors
*
Companion matrix In linear algebra, the Frobenius companion matrix of the monic polynomial
:
p(t)=c_0 + c_1 t + \cdots + c_t^ + t^n ~,
is the square matrix defined as
:C(p)=\begin
0 & 0 & \dots & 0 & -c_0 \\
1 & 0 & \dots & 0 & -c_1 \\
0 & 1 & \dots & 0 & -c_2 ...
*
Faddeev–LeVerrier algorithm
In mathematics (linear algebra), the Faddeev–LeVerrier algorithm is a recursive method to calculate the coefficients of the characteristic polynomial p_A(\lambda)=\det (\lambda I_n - A) of a square matrix, , named after Dmitry Konstantinovi ...
*
Cayley–Hamilton theorem
In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) satisfies ...
*
Samuelson–Berkowitz algorithm
In mathematics, the Samuelson–Berkowitz algorithm efficiently computes the characteristic polynomial of an n\times n matrix whose entries may be elements of any unital commutative ring. Unlike the Faddeev–LeVerrier algorithm, it performs no d ...
References
* T.S. Blyth & E.F. Robertson (1998) ''Basic Linear Algebra'', p 149, Springer .
* John B. Fraleigh & Raymond A. Beauregard (1990) ''Linear Algebra'' 2nd edition, p 246,
Addison-Wesley
Addison-Wesley is an American publisher of textbooks and computer literature. It is an imprint of Pearson PLC, a global publishing and education company. In addition to publishing books, Addison-Wesley also distributes its technical titles throu ...
.
*
* Werner Greub (1974) ''Linear Algebra'' 4th edition, pp 120–5, Springer, .
* Paul C. Shields (1980) ''Elementary Linear Algebra'' 3rd edition, p 274,
Worth Publishers .
*
Gilbert Strang
William Gilbert Strang (born November 27, 1934), usually known as simply Gilbert Strang or Gil Strang, is an American mathematician, with contributions to finite element theory, the calculus of variations, wavelet analysis and linear algebr ...
(1988) ''Linear Algebra and Its Applications'' 3rd edition, p 246,
Brooks/Cole {{ISBN, 0-15-551005-3 .
Polynomials
Linear algebra
Tensors