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Characteristic equation

In mathematics, the characteristic equation is a polynomial equation that plays a central role in determining eigenvalues of square matrices and solving linear homogeneous ordinary differential equations with constant coefficients.^[1]^[2] In linear algebra, for an n \times n square matrix A, the characteristic equation is given by \det(\lambda I_n - A) = 0, where I_n is the n \times n identity matrix and \lambda is a scalar variable; the roots of this equation are precisely the eigenvalues of A, which describe how the matrix scales vectors in certain directions.^[1] The associated polynomial \det(\lambda I_n - A) is known as the characteristic polynomial, a monic polynomial of degree n whose coefficients are functions of the entries of A, such as the trace and determinant for low dimensions.^[1] This equation underpins key theorems like the spectral theorem for symmetric matrices^[3] and the Cayley-Hamilton theorem^[4], which states that every square matrix satisfies its own characteristic equation. In the context of differential equations, the characteristic equation arises when solving an nth-order linear homogeneous equation a_n y^{(n)} + a_{n-1} y^{(n-1)} + \dots + a_1 y' + a_0 y = 0 by assuming a solution of the form y = e^{rt}, leading to the auxiliary equation a_n r^n + a_{n-1} r^{n-1} + \dots + a_1 r + a_0 = 0.^[2] The roots r of this equation dictate the form of the general solution: distinct real roots yield exponential terms c_i e^{r_i t}, repeated roots introduce polynomial factors like t^k e^{rt}, and complex conjugate roots \alpha \pm \beta i produce oscillatory solutions involving sines and cosines via Euler's formula.^[2] This method is essential for analyzing systems in physics, engineering, and other fields where such equations model damped oscillations or growth processes.^[2]

Introduction

Definition

The characteristic equation is a fundamental polynomial equation in mathematics that arises in the analysis of linear systems, particularly in linear algebra and differential equations. In the context of linear algebra, it is defined as the equation \det(A - \lambda I) = 0, where A is an n \times n square matrix, \lambda is a scalar variable, and I is the n \times n identity matrix; the roots of this equation, known as eigenvalues, reveal essential properties of the matrix such as its spectral decomposition and stability characteristics.^[1] In the realm of ordinary differential equations (ODEs), for a linear homogeneous ODE with constant coefficients of order n, such as y^{(n)} + a_{n-1} y^{(n-1)} + \cdots + a_1 y' + a_0 y = 0, the characteristic equation takes the form r^n + a_{n-1} r^{n-1} + \cdots + a_1 r + a_0 = 0, where the roots r determine the form of the general solution and system behavior like exponential growth or decay.^[2] This equation emerges naturally from the requirement to find non-trivial solutions to linear homogeneous systems, such as A \mathbf{v} = \lambda \mathbf{v} for matrices or assuming exponential solutions y = e^{rt} for ODEs, leading to a polynomial whose degree equals the dimension of the system or the order of the equation.^[5] The polynomial nature ensures that the roots can be analyzed using algebraic tools, providing insights into the system's dynamics without solving the full original problem.^[6] A simple example illustrates this for a $2 \times 2 matrix A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, where the characteristic equation simplifies to \det(A - \lambda I) = \lambda^2 - (a+d)\lambda + (ad - bc) = 0, or equivalently \lambda^2 - \operatorname{tr}(A) \lambda + \det(A) = 0, with the roots indicating the eigenvalues.^[7]

Historical Development

The origins of the characteristic equation can be traced to the early 18th century. Abraham de Moivre first employed exponential trial solutions for second-order linear homogeneous ordinary differential equations with constant coefficients in 1730. Leonhard Euler extended this approach to higher-order equations in 1743, reducing the differential equation to an algebraic polynomial equation in r—the precursor to the modern characteristic equation—whose roots determined the general solution form. This innovation was essential for analyzing oscillatory phenomena and marked an early systematic approach to such equations.^[8] During the 19th century, Joseph-Louis Lagrange and Carl Friedrich Gauss advanced the formalization of related concepts through their studies of quadratic forms and determinants, bridging the equation to algebraic structures. Lagrange's investigations into quadratic forms, particularly in his 1788 Mécanique analytique and earlier works on planetary perturbations around 1782, involved diagonalizing symmetric matrices and using what he called the "secular equation"—a determinant-based polynomial akin to the characteristic equation—to identify principal axes and invariants.^[9] Gauss built on this in his 1801 Disquisitiones Arithmeticae, where he systematized the theory of binary quadratic forms, employing determinant computations to classify forms and their equivalence classes, thereby providing tools that facilitated the later recognition of eigenvalues in finite-dimensional linear algebra.^[10] Augustin-Louis Cauchy played a pivotal role in standardizing the terminology and its application to matrices and differential equations. In his 1829 memoir on secular perturbations in astronomy, Cauchy utilized the equation \det(A - \lambda I) = 0 to prove that symmetric matrices possess real eigenvalues, though without yet naming it. He formally introduced the term "équation caractéristique" in 1840 in his Mémoire sur l'intégration des équations différentielles, applying it to the auxiliary polynomial for solving linear systems of differential equations and linking it explicitly to eigenvalues.^[11] The early 20th century saw extensions to infinite-dimensional settings through David Hilbert's foundational contributions to operator theory. In his 1904 treatise Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen, Hilbert generalized the characteristic equation to linear integral equations, introducing the concepts of "eigenwerte" (eigenvalues) and "eigenfunctions" for compact operators on infinite-dimensional spaces, which formed the basis for spectral theory in Hilbert spaces. This work, building on finite-dimensional precedents, enabled the analysis of continuous systems and profoundly influenced functional analysis.^[11]

Linear Algebra Context

Characteristic Polynomial of a Matrix

The characteristic polynomial of an n \times n matrix A over the complex numbers is defined as the determinant p_A(\lambda) = \det(\lambda I_n - A), where I_n is the n \times n identity matrix and \lambda is a scalar variable.^[12] This definition yields a monic polynomial of degree exactly n, with leading coefficient 1.^[12] Some texts employ the alternative convention \det(A - \lambda I_n), which differs by a factor of (-1)^n but shares the same roots.^[13] The polynomial is derived from the matrix pencil \lambda I_n - A, a parameterized family of matrices whose determinant provides a scalar-valued function of \lambda.^[1] For small n, it can be computed explicitly via cofactor expansion. For instance, expanding along the first row of \lambda I_n - A involves summing signed minors, each of which is itself a determinant of a smaller matrix. Key properties include the form p_A(\lambda) = \lambda^n - (\operatorname{tr} A) \lambda^{n-1} + \cdots + (-1)^n \det A, where the trace \operatorname{tr} A is the sum of the diagonal entries and appears as the negative coefficient of \lambda^{n-1}.^[12] More generally, the coefficients are, up to alternating signs, the elementary symmetric functions of the eigenvalues \lambda_1, \dots, \lambda_n of A: specifically, p_A(\lambda) = \prod_{i=1}^n (\lambda - \lambda_i) = \lambda^n - s_1 \lambda^{n-1} + s_2 \lambda^{n-2} - \cdots + (-1)^n s_n, where s_k is the k-th elementary symmetric sum over the eigenvalues.^[14] The constant term is thus (-1)^n \det A, reflecting the product of the eigenvalues up to sign.^[12] For a $2 \times 2 matrix A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, the characteristic polynomial is p_A(\lambda) = \lambda^2 - (a + d) \lambda + (ad - bc) = \lambda^2 - (\operatorname{tr} A) \lambda + \det A.^[1] For a $3 \times 3 matrix A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}, cofactor expansion yields p_A(\lambda) = \lambda^3 - (\operatorname{tr} A) \lambda^2 + \left( \sum_{i=1}^3 \det A_{ii} \right) \lambda - \det A, where A_{ii} is the $2 \times 2 principal minor excluding row i and column i, and the coefficient of \lambda is the sum of those principal minors.^[1] By the fundamental theorem on eigenvalues, the roots of p_A(\lambda) = 0 (counted with algebraic multiplicity) are precisely the eigenvalues of A.^[12] These roots enable the identification of eigenvalues, which are subsequently used to compute corresponding eigenvectors.^[1]

Eigenvalues and Eigenvectors

In linear algebra, the roots of the characteristic equation of a square matrix A are precisely the eigenvalues \lambda of A, which satisfy the equation A\mathbf{v} = \lambda \mathbf{v} for some nonzero vector \mathbf{v}, known as an eigenvector corresponding to \lambda.^[15]^[16] This relationship arises directly from the definition of the characteristic equation \det(A - \lambda I) = 0, where the solutions \lambda identify the scalars by which A scales its eigenvectors during linear transformations. Geometrically, eigenvectors represent directions unchanged by the transformation except for scaling by \lambda, providing insight into the matrix's action on vector spaces.^[17] Each eigenvalue \lambda has an algebraic multiplicity, defined as the multiplicity of \lambda as a root of the characteristic polynomial, and a geometric multiplicity, which is the dimension of the corresponding eigenspace (the null space of A - \lambda I).^[18] The geometric multiplicity is always less than or equal to the algebraic multiplicity, and equality holds for all eigenvalues if and only if the matrix is diagonalizable.^[19] A matrix is defective if, for some eigenvalue, the geometric multiplicity is strictly less than the algebraic multiplicity, as in the example of the matrix \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, where \lambda = 1 has algebraic multiplicity 2 but geometric multiplicity 1, yielding only one independent eigenvector.^[18] A matrix A is diagonalizable if it possesses a full set of n linearly independent eigenvectors for an n \times n matrix, allowing a similarity transformation P^{-1}AP = D where P has the eigenvectors as columns and D is a diagonal matrix with the eigenvalues on the diagonal.^[20] This decomposition simplifies computations like matrix powers, as A^k = PDP^{-1} implies A^k = PD^kP^{-1}.^[21] Distinct eigenvalues guarantee linear independence of eigenvectors, but repeated eigenvalues may or may not, depending on the eigenspace dimension.^[20] For instance, consider the 2×2 rotation matrix R = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} for \theta \neq 0, \pi, which has characteristic equation roots \lambda = e^{\pm i\theta}, complex eigenvalues with no real eigenvectors since the real eigenspaces are trivial.^[22] Geometrically, this reflects the matrix's pure rotation without scaling or fixed directions in the real plane.^[17] When a matrix is not diagonalizable, its Jordan canonical form provides a block-diagonal representation with Jordan blocks featuring the eigenvalue on the diagonal and 1's on the superdiagonal, capturing the structure of generalized eigenvectors for defective cases.^[23] This form generalizes diagonalization while preserving the eigenvalues.^[24]

Differential Equations Context

For Linear Homogeneous ODEs with Constant Coefficients

In the context of ordinary differential equations, the characteristic equation is employed to solve linear homogeneous equations with constant coefficients. The general form of an nth-order such equation is

a_n y^{(n)}(x) + a_{n-1} y^{(n-1)}(x) + \cdots + a_1 y'(x) + a_0 y(x) = 0,

where the coefficients a_k (for k = 0, 1, \dots, n) are real constants with a_n \neq 0.^[25] To find solutions, an exponential trial function y(x) = e^{rx} is assumed, where r is a constant parameter. The derivatives follow as y^{(k)}(x) = r^k e^{rx} for each order k. Substituting these into the ODE yields

e^{rx} \left( a_n r^n + a_{n-1} r^{n-1} + \cdots + a_1 r + a_0 \right) = 0.

Since e^{rx} \neq 0 for all x, the equation simplifies to the characteristic equation

a_n r^n + a_{n-1} r^{n-1} + \cdots + a_1 r + a_0 = 0,

a polynomial of degree n in r.^[26]^[25] The roots of this characteristic equation determine the structure of the solutions to the original ODE. Given real coefficients, the roots are either real numbers or complex numbers appearing in conjugate pairs. Real roots may be distinct or have multiplicities greater than one (repeated roots), while complex roots take the form \alpha \pm i\beta with \beta \neq 0. For higher-order equations, multiplicities can exceed two, affecting the solution basis accordingly.^[26]^[2]^[25] As an example, consider the second-order equation y''(x) + 3y'(x) + 2y(x) = 0. The corresponding characteristic equation is r^2 + 3r + 2 = 0, which factors as (r + 1)(r + 2) = 0, giving roots r = -1 and r = -2, both real and distinct.^[2]

Solution via Characteristic Roots

Once the roots of the characteristic equation are determined for an nth-order linear homogeneous ordinary differential equation (ODE) with constant coefficients, the general solution is constructed as a linear combination of basis functions derived from those roots. This approach leverages the fact that exponential functions e^{rt} form solutions corresponding to each root r, with modifications for multiplicities and complex values to ensure linear independence. The arbitrary constants in the linear combination are then fixed by initial or boundary conditions to yield a unique solution.^[2] For distinct real roots r_1, r_2, \dots, r_n, the general solution takes the form

y(t) = c_1 e^{r_1 t} + c_2 e^{r_2 t} + \dots + c_n e^{r_n t},

where c_1, c_2, \dots, c_n are arbitrary constants determined by initial conditions. This structure arises because each distinct root produces a linearly independent exponential solution.^[27]^[2] When a real root r has multiplicity m > 1, the corresponding basis functions include polynomial factors to maintain linear independence, yielding terms t^k e^{rt} for k = 0, 1, \dots, m-1. The contribution to the general solution from this repeated root is thus e^{rt} (c_1 + c_2 t + \dots + c_m t^{m-1}), combined linearly with solutions from other roots. For instance, consider the second-order ODE y'' - 2y' + y = 0, whose characteristic equation r^2 - 2r + 1 = 0 has a repeated root r = 1. The general solution is y(t) = (c_1 + c_2 t) e^t.^[27]^[2] For complex conjugate roots \alpha \pm \beta i (with \beta \neq 0), the solutions are expressed in real form using Euler's formula, producing oscillatory components modulated by exponential decay or growth. The pair contributes e^{\alpha t} (c_1 \cos(\beta t) + c_2 \sin(\beta t)) to the general solution, ensuring real-valued functions. If such a pair has multiplicity greater than one, additional factors of t^k (for k = 0 to the multiplicity minus one) are included for each trigonometric term. Applying initial conditions to the full linear combination determines the specific solution satisfying the problem constraints.^[27]^[2]

Other Mathematical Contexts

In Control Theory

In control theory, the characteristic equation is fundamental to analyzing the stability and dynamic performance of linear time-invariant (LTI) systems, as its roots determine the locations of the system's poles in the complex plane, which govern the transient response and asymptotic behavior. For systems modeled in state-space form as \dot{x} = Ax + Bu and y = Cx + Du, the characteristic equation arises from the determinant condition \det(sI - A) = 0, where A is the system matrix and s is the complex variable; the roots of this equation are the eigenvalues of A, representing the natural frequencies and damping characteristics of the system's modes.^[28] This formulation directly links the internal dynamics to external inputs and outputs, enabling the design of controllers that place these roots to achieve desired stability and response specifications. The roots of the state-space characteristic equation correspond precisely to the poles of the system's transfer function G(s) = C(sI - A)^{-1}B + D, which describe the input-output relationship in the Laplace domain; these poles dictate the system's stability, as all must lie in the open left-half s-plane for asymptotic stability.^[29] Stability analysis often employs the Routh-Hurwitz criterion, which provides necessary and sufficient conditions on the coefficients of the characteristic polynomial to ensure all roots have negative real parts without explicitly solving for them; for a quadratic characteristic equation s^2 + as + b = 0, the system is stable if and only if a > 0 and b > 0, corresponding to positive damping and stiffness terms in second-order systems.^[30] This criterion extends to higher-order polynomials via the Routh array, facilitating rapid assessment in control design.^[31] A practical application appears in feedback control systems, where the characteristic equation is derived from the closed-loop transfer function and analyzed using the root locus technique to visualize pole migration as a gain parameter varies. For instance, consider a unity-feedback system with open-loop transfer function G(s) = K \frac{s+1}{s^2 + 2s + 1}, yielding the characteristic equation $1 + K \frac{s+1}{s^2 + 2s + 1} = 0; the root locus begins at the open-loop poles—a double pole at s = -1—and as K increases from 0, one branch remains fixed at s = -1, while the other moves leftward along the real axis toward negative infinity, revealing regions of stability for K > 0 and aiding in selecting K for optimal damping and settling time. This method highlights how gain adjustments influence performance metrics like overshoot and bandwidth without repeated eigenvalue computations.

In Difference Equations

In linear homogeneous difference equations with constant coefficients, the characteristic equation plays a central role in determining the general solution, analogous to its role in ordinary differential equations but adapted to discrete-time systems. These equations take the form a_n y_{k+n} + a_{n-1} y_{k+n-1} + \cdots + a_1 y_{k+1} + a_0 y_k = 0, where a_i are constants and n is the order of the recurrence. To solve such an equation, one assumes a solution of the form y_k = r^k, where r is a constant to be determined. Substituting this assumed form into the equation yields the characteristic equation: a_n r^n + a_{n-1} r^{n-1} + \cdots + a_1 r + a_0 = 0. The roots of this characteristic equation determine the form of the general solution. For an n-th order equation with n distinct roots r_1, r_2, \dots, r_n, the general solution is y_k = c_1 r_1^k + c_2 r_2^k + \cdots + c_n r_n^k, where the c_i are arbitrary constants determined by initial conditions. If there are repeated roots, say a root r with multiplicity m, the corresponding terms in the solution include polynomial factors: (c_1 + c_2 k + \cdots + c_m k^{m-1}) r^k. For complex roots, which occur in conjugate pairs for equations with real coefficients, the solution can be expressed using polar form: if r = \rho e^{\pm i \theta}, then the terms are \rho^k (A \cos(k \theta) + B \sin(k \theta)). A simple example illustrates this approach. Consider the second-order equation y_{k+2} - 3 y_{k+1} + 2 y_k = 0. The characteristic equation is r^2 - 3r + 2 = 0, which factors as (r-1)(r-2) = 0, yielding distinct roots r=1 and r=2. Thus, the general solution is y_k = c_1 \cdot 1^k + c_2 \cdot 2^k = c_1 + c_2 2^k. In the context of the z-transform, which is the discrete analog of the Laplace transform, the roots of the characteristic equation correspond to the poles of the transfer function for the system described by the difference equation. This connection is fundamental in signal processing and control theory for analyzing stability and response in discrete-time systems.

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