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Characteristic polynomial

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Polynomial whose roots are the eigenvalues of a matrix

This article is about the characteristic polynomial of a matrix or of an endomorphism of vector spaces. For the characteristic polynomial of a matroid, seeMatroid. For that of a graded poset, seeGraded poset.

Inlinear algebra, thecharacteristic polynomial of asquare matrix is apolynomial which isinvariant undermatrix similarity and has theeigenvalues asroots. It has thedeterminant and thetrace of the matrix among its coefficients. Thecharacteristic polynomial of anendomorphism of a finite-dimensionalvector space is the characteristic polynomial of the matrix of that endomorphism over any basis (that is, the characteristic polynomial does not depend on the choice of abasis). Thecharacteristic equation, also known as thedeterminantal equation,^[1]^[2]^[3] is the equation obtained by equating the characteristic polynomial to zero.

Inspectral graph theory, thecharacteristic polynomial of agraph is the characteristic polynomial of itsadjacency matrix.^[4]

Motivation

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Eigenvalues and eigenvectors play a fundamental role inlinear algebra, since, given alinear 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, suppose the transformation is represented by a square matrix $A . {\displaystyle A.}$ Then an eigenvector $\mathbf {v}$ and the corresponding eigenvalue $\lambda$ must satisfy the equation $A\mathbf {v} =\lambda \mathbf {v} ,$ or, equivalently (since $\lambda \mathbf {v} =\lambda I\mathbf {v}$ ), $(\lambda I-A)\mathbf {v} =\mathbf {0}$ where $I {\displaystyle I}$ is theidentity matrix, and $\mathbf {v} \neq \mathbf {0}$ (although the zero vector satisfies this equation for every $\lambda ,$ it is not considered an eigenvector).

It follows that the matrix $(\lambda I-A)$ must besingular, and its determinant $\det(\lambda I-A)=0$ must be zero.

In other words, the eigenvalues ofA are theroots of $\det(xI-A),$ which is amonic polynomial inx of degreen ifA is an×n matrix. This polynomial is thecharacteristic polynomial ofA.

Formal definition

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Consider an $n\times n$ matrix $A . {\displaystyle A.}$ The characteristic polynomial of $A, {\displaystyle A,}$ denoted by $p_{A}(t),$ is the polynomial defined by^[5] $p_{A}(t)=\det(tI-A)$ where $I {\displaystyle I}$ denotes the $n\times n$ identity matrix.

Some authors define the characteristic polynomial to be $\det(A-tI).$ 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 $A {\displaystyle A}$ ; however the definition above always gives amonic polynomial, whereas the alternative definition is monic only when $n {\displaystyle n}$ is even.

Examples

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To compute the characteristic polynomial of the matrix $A={\begin{pmatrix}2&1\\-1&0\end{pmatrix}}.$ thedeterminant of the following is computed: $tI-A={\begin{pmatrix}t-2&-1\\1&t-0\end{pmatrix}}$ and found to be $(t-2)t-1(-1)=t^{2}-2t+1\,\!,$ the characteristic polynomial of $A . {\displaystyle A.}$

Another example useshyperbolic functions of ahyperbolic angle φ.For the matrix take $A={\begin{pmatrix}\cosh(\varphi )&\sinh(\varphi )\\\sinh(\varphi )&\cosh(\varphi )\end{pmatrix}}.$ Its characteristic polynomial is $\det(tI-A)=(t-\cosh(\varphi ))^{2}-\sinh ^{2}(\varphi )=t^{2}-2t\ \cosh(\varphi )+1=(t-e^{\varphi })(t-e^{-\varphi }).$

Properties

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The characteristic polynomial $p_{A}(t)$ of a $n\times n$ matrix $A {\displaystyle A}$ ismonic (its leading coefficient is $1 {\displaystyle 1}$ ) and its degree is $n . {\displaystyle n.}$ The most important fact about the characteristic polynomial was already mentioned in themotivational paragraph: the eigenvalues of $A {\displaystyle A}$ are precisely theroots of $p_{A}(t)$ (this also holds for theminimal polynomial of $A, {\displaystyle A,}$ but its degree may be less than $n {\displaystyle n}$ ). All coefficients of the characteristic polynomial arepolynomial expressions in the entries of the matrix. In particular its constant coefficient of $t^{0}$ is $\det(-A)=(-1)^{n}\det(A),$ the coefficient of $t^{n}$ is 1, and the coefficient of $t^{n-1}$ istr(−A) = −tr(A), wheretr(A) is thetrace of $A . {\displaystyle A.}$ (The signs given here correspond to theformal definition given in the previous section; for the alternative definition these would instead be $\det(A)$ and(−1)^{n – 1}tr(A) respectively.^[6])

For a $2\times 2$ matrix $A, {\displaystyle A,}$ the characteristic polynomial is thus given by $t^{2}-\operatorname {tr} (A)t+\det(A).$

Using the language ofexterior algebra, the characteristic polynomial of an $n\times n$ matrix $A {\displaystyle A}$ may be expressed as $p_{A}(t)=\sum _{k=0}^{n}t^{n-k}(-1)^{k}\operatorname {tr} \left(\textstyle \bigwedge ^{k}A\right)$ where ${\textstyle \operatorname {tr} \left(\bigwedge ^{k}A\right)}$ is thetrace of the $k {\displaystyle k}$ thexterior power of $A, {\displaystyle A,}$ which has dimension ${\textstyle {\binom {n}{k}}.}$ This trace may be computed as the sum of allprincipal minors of $A {\displaystyle A}$ of size $k . {\displaystyle k.}$ The recursiveFaddeev–LeVerrier algorithm computes these coefficients more efficiently^{[clarification needed]}.

TheCayley–Hamilton theorem states that replacing $t {\displaystyle t}$ by $A {\displaystyle A}$ in the characteristic polynomial (interpreting the resulting powers as matrix powers, and the constant term $c {\displaystyle c}$ as $c {\displaystyle c}$ times the identity matrix) yields thezero matrix. Informally speaking, every matrix satisfies its own characteristic equation. This statement is equivalent to saying that theminimal polynomial of $A {\displaystyle A}$ divides the characteristic polynomial of $A . {\displaystyle A.}$

Twosimilar 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 $A {\displaystyle A}$ and itstranspose have the same characteristic polynomial. $A {\displaystyle A}$ is similar to atriangular matrix if and only if its characteristic polynomial can be completely factored into linear factors over $K {\displaystyle K}$ (the same is true with the minimal polynomial instead of the characteristic polynomial). In this case $A {\displaystyle A}$ is similar to a matrix inJordan normal form.

Characteristic polynomial of a product of two matrices

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If $A {\displaystyle A}$ and $B {\displaystyle B}$ are two square $n\times n$ matrices then characteristic polynomials of $A B {\displaystyle AB}$ and $B A {\displaystyle BA}$ coincide: $p_{AB}(t)=p_{BA}(t).\,$

Proof: If $\lambda$ is a non-zero generalized eigenvalue of $A B {\displaystyle AB}$ of algebraic multiplicity $k {\displaystyle k}$ , and $v {\displaystyle v}$ belongs to the kernel of $(BA-\lambda )^{k}$ , then $A v {\displaystyle Av}$ belongs to the kernel of $(AB-\lambda )^{k}$ , so the non-zero generalized eigenspaces of $A B {\displaystyle AB}$ and $B A {\displaystyle BA}$ have the same dimension. Therefore, since $A B {\displaystyle AB}$ and $B A {\displaystyle BA}$ are both $n\times n$ , the remaining generalized eigenspaces, with eigenvalue 0, have the same dimension. Therefore $A B {\displaystyle AB}$ and $B A {\displaystyle BA}$ have the same characteristic polynomial, because all generalized eigenvalues are the same, with the same algebraic multiplicities.

More generally, if $A {\displaystyle A}$ is a matrix of order $m\times n$ and $B {\displaystyle B}$ is a matrix of order $n\times m,$ then $A B {\displaystyle AB}$ is $m\times m$ and $B A {\displaystyle BA}$ is $n\times n$ matrix, and one has $p_{BA}(t)=t^{n-m}p_{AB}(t).\,$

To prove this, one may suppose $n>m,$ by exchanging, if needed, $A {\displaystyle A}$ and $B . {\displaystyle B.}$ Then, by bordering $A {\displaystyle A}$ on the bottom by $n-m$ rows of zeros, and $B {\displaystyle B}$ on the right, by, $n-m$ columns of zeros, one gets two $n\times n$ matrices $A^{\prime }$ and $B^{\prime }$ such that $B^{\prime }A^{\prime }=BA$ and $A^{\prime }B^{\prime }$ is equal to $A B {\displaystyle AB}$ bordered by $n-m$ rows and columns of zeros. The result follows from the case of square matrices, by comparing the characteristic polynomials of $A^{\prime }B^{\prime }$ and $A B . {\displaystyle AB.}$

Characteristic polynomial ofA^k

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If $\lambda$ is an eigenvalue of a square matrix $A {\displaystyle A}$ with eigenvector $\mathbf {v} ,$ then $\lambda ^{k}$ is an eigenvalue of $A^{k}$ because $A^{k}{\textbf {v}}=A^{k-1}A{\textbf {v}}=\lambda A^{k-1}{\textbf {v}}=\dots =\lambda ^{k}{\textbf {v}}.$

The multiplicities can be shown to agree as well, and this generalizes to any polynomial in place of $x^{k}$ :^[7]

Theorem— Let $A {\displaystyle A}$ be a square $n\times n$ matrix and let $f(t)$ be a polynomial. If the characteristic polynomial of $A {\displaystyle A}$ has a factorization $p_{A}(t)=(t-\lambda _{1})(t-\lambda _{2})\cdots (t-\lambda _{n})$ then the characteristic polynomial of the matrix $f(A)$ is given by $p_{f(A)}(t)=(t-f(\lambda _{1}))(t-f(\lambda _{2}))\cdots (t-f(\lambda _{n})).$

That is, the algebraic multiplicity of $\lambda$ in $f(A)$ equals the sum of algebraic multiplicities of $\lambda '$ in $A {\displaystyle A}$ over $\lambda '$ such that $f(\lambda ')=\lambda .$ In particular, $\operatorname {tr} (f(A))=\textstyle \sum _{i=1}^{n}f(\lambda _{i})$ and $\operatorname {det} (f(A))=\textstyle \prod _{i=1}^{n}f(\lambda _{i}).$ Here a polynomial $f(t)=t^{3}+1,$ for example, is evaluated on a matrix $A {\displaystyle A}$ simply as $f(A)=A^{3}+I.$

The theorem applies to matrices and polynomials over any field orcommutative ring.^[8]However, the assumption that $p_{A}(t)$ has a factorization into linear factors is not always true, unless the matrix is over analgebraically 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 $p_{A}(t)=\left(t-\lambda _{1}\right)\left(t-\lambda _{2}\right)\cdots \left(t-\lambda _{n}\right)$ where $\lambda _{1},\lambda _{2},\ldots ,\lambda _{n}$ are the eigenvalues of $A, {\displaystyle A,}$ possibly repeated. Moreover, theJordan decomposition theorem guarantees that any square matrix $A {\displaystyle A}$ can be decomposed as $A=S^{-1}US,$ where $S {\displaystyle S}$ is aninvertible matrix and $U {\displaystyle U}$ isupper triangularwith $\lambda _{1},\ldots ,\lambda _{n}$ on the diagonal (with each eigenvalue repeated according to its algebraic multiplicity).(The Jordan normal form has stronger properties, but these are sufficient; alternatively theSchur decomposition can be used, which is less popular but somewhat easier to prove).

Let ${\textstyle f(t)=\sum _{i}\alpha _{i}t^{i}.}$ Then $f(A)=\textstyle \sum \alpha _{i}(S^{-1}US)^{i}=\textstyle \sum \alpha _{i}S^{-1}USS^{-1}US\cdots S^{-1}US=\textstyle \sum \alpha _{i}S^{-1}U^{i}S=S^{-1}(\textstyle \sum \alpha _{i}U^{i})S=S^{-1}f(U)S.$ For an upper triangular matrix $U {\displaystyle U}$ with diagonal $\lambda _{1},\dots ,\lambda _{n},$ the matrix $U^{i}$ is upper triangular with diagonal $\lambda _{1}^{i},\dots ,\lambda _{n}^{i}$ in $U^{i},$ and hence $f(U)$ is upper triangular with diagonal $f\left(\lambda _{1}\right),\dots ,f\left(\lambda _{n}\right).$ Therefore, the eigenvalues of $f(U)$ are $f(\lambda _{1}),\dots ,f(\lambda _{n}).$ Since $f(A)=S^{-1}f(U)S$ issimilar to $f(U),$ it has the same eigenvalues, with the same algebraic multiplicities.

Secular function and secular equation

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Secular function

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The termsecular function has been used for what is now calledcharacteristic polynomial (in some literature the term secular function is still used). The term comes from the fact that the characteristic polynomial was used to calculatesecular perturbations (on a time scale of a century, that is, slow compared to annual motion) of planetary orbits, according toLagrange's theory of oscillations.

Secular equation

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Secular equation may have several meanings.

Inlinear algebra it is sometimes used in place of characteristic equation.
Inastronomy 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.^[9]

Inmolecular orbital calculations relating to the energy of the electron and itswave function it is also used instead of the characteristic equation.

For general associative algebras

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The above definition of the characteristic polynomial of a matrix $A\in M_{n}(F)$ with entries in a field $F {\displaystyle F}$ generalizes without any changes to the case when $F {\displaystyle F}$ is just acommutative ring.Garibaldi (2004) defines the characteristic polynomial for elements of an arbitrary finite-dimensional (associative, but not necessarily commutative) algebra over a field $F {\displaystyle F}$ and proves the standard properties of the characteristic polynomial in this generality.

References

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^Guillemin, Ernst (1953).Introductory Circuit Theory. Wiley. pp. 366, 541.ISBN 0471330663.{{cite book}}:ISBN / Date incompatibility (help)
^Forsythe, George E.; Motzkin, Theodore (January 1952)."An Extension of Gauss' Transformation for Improving the Condition of Systems of Linear Equations"(PDF).Mathematics of Computation.6 (37):18–34.doi:10.1090/S0025-5718-1952-0048162-0. Retrieved3 October 2020.
^Frank, Evelyn (1946)."On the zeros of polynomials with complex coefficients"(PDF).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". RetrievedAugust 26, 2011.
^Steven Roman (1992).Advanced linear algebra (2 ed.). Springer. p. 137.ISBN 3540978372.
^Theorem 4 in theselecture 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". RetrievedJanuary 21, 2010.

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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
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