How to write the characteristic equation
The characteristic equation, which is calculated on the basis of primarily the eigenvalues (values) have found wide application in mathematics, physics and engineering. They can be found in the decisions of the automatic control problems, solutions of differential equations, and so on. N.
Instruction write the characteristic equation
To answer to the question should be approached by considering the simplest tasks, for which the characteristic equation may be required. First of all - this is a normal solution of the homogeneous system of homogeneous differential equations (Linear Ordinary Differential Equations). Its form is shown in Figure 1.Uchityvaya notation shown in Fig. 1. Rewrite the system matrix vide.Poluchite Y '= AY.
It is known that a fundamental system of solutions (FSS), the problem under consideration is in the form of Y = exp [kx] B, where B - column constant. Then Y '= kY. There is a system AY-kEY = 0 (E - the identity matrix). Or (A-kE) Y = 0. We wanted to find a non-zero solutions, so the system of homogeneous equations has a degenerate matrix and, consequently, the determinant of a matrix is zero. In expanded form the determinant (see. Fig. 2) .On Fig. 2 as a determinant of recorded algebraic equation of n-th order and his decision to allow the SDF to make the original system. This equation is called the characteristic.
Now consider the Linear Ordinary Differential Equations of n-th order (see. Fig. 3) .If the left part of it designated as a linear differential operator L [y], the Linear Ordinary Differential Equations rewritten in the form L [y] = 0. If seek solutions Linear Ordinary Differential Equations as y = exp (kx), then y '= kexp (kx), y' '= (k ^ 2) exp (kx), ..., y ^ (n-1) = (k ^ ( n-1)) exp (kx), y ^ n = (k ^ n) exp (kx) and, after reduction at y = exp (kx), we obtain the equation: k ^ n + (a1) k ^ (n-1 ) + ... + a (n-1) k + an = 0, which is also called the characteristic.
In order to ensure that the essence of the latter characteristic equation remains the same (that is, it is not some other object), click Linear Ordinary Differential Equations of the n-th order to the normal system of Linear Ordinary Differential Equations by successive substitutions. The first of these y1 = y, and daleey1 '= y2, y2'1 = y3, ..., y (n-1)' = yn, yn '= - an * y1-a (n-2) * yn- ... - a1 * y (n-1).
Record emerged system, make it the characteristic equation in the form of a determinant, open it and make sure that we obtain the characteristic equation for the Linear Ordinary Differential Equations of n-th order. At the same time there and the statement about the fundamental meaning of the characteristic equation.
Go to the general problem of finding the eigenvalues of the linear transformations (they can be and differential), which comprises the step of drawing up the characteristic equation. The number k is called the characteristic value (number) of the linear transformation A if there exists a vector x such that Ax = kx.Poskolku each linear transformation matrix it can be delivered clearly, the problem reduces to the compilation of the characteristic equation for a square matrix. This is done exactly as in the initial example for normal systems Linear Ordinary Differential Equations. Simply replace the symbols on the x-y, if after recording the characteristic equation will follow some other action. If not, then it should not do. Just take a matrix A (see. Fig. 1) and record the answer in the form of a determinant (see. Figure 2). After disclosure of the determinant work is completed.