Search results
Results From The WOW.Com Content Network
If a second-order differential equation has a characteristic equation with complex conjugate roots of the form r 1 = a + bi and r 2 = a − bi, then the general solution is accordingly y(x) = c 1 e (a + bi )x + c 2 e (a − bi )x. By Euler's formula, which states that e iθ = cos θ + i sin θ, this solution can be rewritten as follows:
The characteristic equation, also known as the determinantal equation, [1] [2] [3] is the equation obtained by equating the characteristic polynomial to zero. In spectral graph theory , the characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix .
Characteristics are also a powerful tool for gaining qualitative insight into a PDE. One can use the crossings of the characteristics to find shock waves for potential flow in a compressible fluid. Intuitively, we can think of each characteristic line implying a solution to along itself. Thus, when two characteristics cross, the function ...
This polynomial is called the characteristic polynomial of A. Equation is called the characteristic equation or the secular equation of A. The fundamental theorem of algebra implies that the characteristic polynomial of an n-by-n matrix A, being a polynomial of degree n, can be factored into the product of n linear terms,
Characteristic equation may refer to: Characteristic equation (calculus), used to solve linear differential equations; Characteristic equation, the equation obtained by equating to zero the characteristic polynomial of a matrix or of a linear mapping; Method of characteristics, a technique for solving partial differential equations
If, on the other hand, we know the characteristic function φ and want to find the corresponding distribution function, then one of the following inversion theorems can be used. Theorem. If the characteristic function φ X of a random variable X is integrable, then F X is absolutely continuous, and therefore X has a probability density function.
The importance of the criterion is that the roots p of the characteristic equation of a linear system with negative real parts represent solutions e pt of the system that are stable . Thus the criterion provides a way to determine if the equations of motion of a linear system have only stable solutions, without solving the system directly.
The closed-loop poles, or eigenvalues, are obtained by solving the characteristic equation + =. In general, the solution will be n complex numbers where n is the order of the characteristic polynomial. The preceding is valid for single-input-single-output systems (SISO).