# Characteristic Equation

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## characteristic equation

[‚kar·ik·tə′ris·tik i′kwā·zhən]*A*

**u**= λ

**u**, which can have a solution only when the parameter λ has certain values, where

*A*can be a square matrix which multiplies the vector

**u**, or a linear differential or integral operator which operates on the function

**u**, or in general, any linear operator operating on the vector

**u**in a finite or infinite dimensional vector space. Also known as eigenvalue equation.

*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.

## Characteristic Equation

in mathematics. **(1)** The characteristic equation of a matrix is the algebraic equation

The determinant on the left-hand side of the characteristic equation is obtained by subtracting λ from the diagonal elements of the matrix . This determinant is a polynomial in λ and is called the characteristic polynomial.

The explicit form of the equation is (–λ)^{n} + *S*_{1}(–λ)^{n – 1} + *S*_{2}(–λ)^{n – 2} + · · · + *S _{n}* = 0. Here,

*S*

_{1}=

*a*

_{11}+

*a*

_{22}+ · · · +

*a*is the trace of the matrix;

_{nn}*S*

_{2}, is the sum of all minors of order two, that is, of all minors

where *i* < *k*; *S*_{3} through *S*_{n – 1} are defined correspondingly; and *S _{n}* is the determinant of the matrix

*A*.

The roots λ_{1}, λ_{2}, . . ., λ_{n} of the characteristic equation are called the eigenvalues of *A*. If *A* is real symmetric (more generally, Hermitian symmetric), then the λ_{k} are real. If *A* is real and skew symmetric, then the λ_{k} are pure imaginary. If *A* is orthogonal (more generally, unitary), then all | λ_{k}| = 1.

Characteristic equations are encountered in a great many areas of mathematics, physics, mechanics, and engineering. Because of their application in astronomy to the determination of secular perturbations of planets, they are also called secular equations.

**(2)** The characteristic equation of a linear differential equation with constant coefficients *a*_{0}*y*^{(n)} + *a*_{1}*y*^{(n – 1)} + · · · + *a*_{(n – 1)}*y′* + *a _{n}y* = 0 is the algebraic equation obtained from this differential equation by replacing

*y*and its derivatives by suitable powers of λ, that is, the equation

*a*

_{0}λ

^{n}+

*a*

_{1}λ

^{n – 1}+ · · · +

*a*

_{n – 1}λ +

*a*= 0. We are led to this equation if we look for solutions of the given differential equation that have the form

_{n}*y*=

*ce*

^{λx}. For a system of linear differential equations

the characteristic equation is

and thus coincides with the characteristic equation of the matrix whose elements are the coefficients of the equations of the system.