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 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, also known as the determinantal equation,[1][2][3] is the equation obtained by equating to zero the characteristic polynomial.

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

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 a₁, a₂, a₃, etc. then the characteristic polynomial will be:

(t-a_{1})(t-a_{2})(t-a_{3})\cdots .

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 a nonzero vector v, called an eigenvector, such that

A\mathbf {v} =\lambda \mathbf {v} ,

or, equivalently,

(\lambda I-A)\mathbf {v} =0

(where $I$ is the identity matrix). Since $v$ must be non-zero, this means that the matrix $λI - A$ has a nonzero kernel. Thus this matrix is not invertible, and the same is true for its determinant, which must therefore be zero. Thus the eigenvalues of $A$ are the roots of $det(λI - A)$ , which 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[5]

p_{A}(t)=\det \left(tI-A\right)

where I denotes the n×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; however the definition above always gives a monic polynomial, whereas the alternative definition is monic only when n is even.

Examples

Suppose we want to compute the characteristic polynomial of the matrix

A={\begin{pmatrix}2&1\\-1&0\end{pmatrix}}.

We now compute the determinant of

tI-A={\begin{pmatrix}t-2&-1\\1&t-0\end{pmatrix}}

which is $(t-2)t-1(-1)=t^{2}-2t+1\,\!,$ the characteristic polynomial of A.

Another example uses hyperbolic functions of a hyperbolic 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

The characteristic polynomial $p A (t)$ of a $n \times n$ matrix 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 $n$ ). All coefficients of the characteristic polynomial are 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 trace of $A$ . (The signs given here correspond to the formal definition given in the previous section;[6] for the alternative definition these would instead be $det(A)$ and $(-1) n - 1 tr(A)$ respectively.[7])

For a 2×2 matrix A, the characteristic polynomial is thus given by

t^{2}-\operatorname {tr} (A)t+\det(A).

Using the language of exterior algebra, one may compactly express the characteristic polynomial of an n×n matrix A as

p_{A}(t)=\sum _{k=0}^{n}t^{n-k}(-1)^{k}\operatorname {tr} (\Lambda ^{k}A)

where tr(Λ^kA) is the trace of the k^th exterior power of A, which has dimension ${\tbinom {n}{k}}$ . 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 of the field of the coefficients is 0, each such trace may alternatively be computed as a single determinant, that of the $k \times k$ matrix,

\operatorname {tr} (\Lambda ^{k}A)={\frac {1}{k!}}{\begin{vmatrix}\operatorname {tr} A&k-1&0&\cdots &\\\operatorname {tr} A^{2}&\operatorname {tr} A&k-2&\cdots &\\\vdots &\vdots &&\ddots &\vdots \\\operatorname {tr} A^{k-1}&\operatorname {tr} A^{k-2}&&\cdots &1\\\operatorname {tr} A^{k}&\operatorname {tr} A^{k-1}&&\cdots &\operatorname {tr} A\end{vmatrix}}~.

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:

p_{AB}(t)=p_{BA}(t).\,

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

BA=A^{-1}(AB)A.

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

p_{BA}(t)=t^{n-m}p_{AB}(t).\,

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.

Characteristic polynomial of A^k

If $\lambda$ is an eigenvalue of a square matrix A with eigenvector v, then clearly $\lambda ^{k}$ is an eigenvalue of A^k

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 $x^{k}$ :[8]

Theorem — Let A be a square n×n matrix and let $f(t)$ be a polynomial. If the characteristic polynomial of 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 $A$ 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 simply as $f(A)=A^{3}+1$ .

The theorem applies to matrices and polynomials over any field or commutative ring.[9] However, the assumption that $p_{A}(t)$ 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

p_{A}(t)=(t-\lambda _{1})(t-\lambda _{2})\cdots (t-\lambda _{n})

where $\lambda _{1},\lambda _{2},\dots ,\lambda _{n}$ are the eigenvalues of $A$ , possibly repeated. Moreover, the Jordan decomposition theorem guarantees that any square matrix $A$ can be decomposed as $A=S^{-1}US$ , where $S$ is an invertible matrix and $U$ is upper triangular with $\lambda _{1},\dots ,\lambda _{n}$ on the diagonal (with each eigenvalue repeated according to its algebraic multiplicity). (The Jordan normal form has stronger properties, but these are sufficient; alternatively the Schur decomposition can be used, which is less popular but somewhat easier to prove).

Let $f(t)=\textstyle \sum _{i}\alpha _{i}t^{i}$ . 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

.

It is easy to check for an upper triangular matrix $U$ 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 $f(\lambda _{1}),\dots ,f(\lambda _{n})$ . Therefore, the eigenvalues of $f(U)$ are $f(\lambda _{1}),\dots ,f(\lambda _{n})$ . Since $f(A)=S^{-1}f(U)S$ is similar to $f(U)$ , it has the same eigenvalues, with the same algebraic multiplicities.

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.[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 algebras

The above definition of the characteristic polynomial of a matrix $A\in M_{n}(F)$ with entries in a field F generalizes without any changes to the case when F 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 F and proves the standard properties of the characteristic polynomial in this generality.

References

Guillemin, Ernst (1953). Introductory Circuit Theory. Wiley. pp. 366, 541. ISBN 0471330663. Lay summary.
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. Retrieved 3 October 2020. Lay summary.
"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
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 .

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[1] Guillemin, Ernst (1953). Introductory Circuit Theory. Wiley. pp. 366, 541. ISBN 0471330663. Lay summary.

[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. Retrieved 3 October 2020. Lay summary.

[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

[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.