Two linearly independent solutions for this ODE can be straightforwardly found using characteristic equations except for the case when the discriminant, b ^ 2-4 a c, vanishes.
32.
Thus, the closed-loop poles of the closed-loop transfer function are the roots of the characteristic equation 1 + G ( s ) H ( s ) = 0.
33.
Just as in the theory of the eigenvectors of an ordinary linear operator, the eigenbivectors of the Weyl tensor can occur with various characteristic equation, in this case a quartic equation.
34.
To simplify these characteristic equations we can make the transformations such that \ frac { dr } { dt } = l _ i \ frac { du _ i } { dt}
35.
As A . J . Coleman says, " He exhibited the characteristic equation of the Weyl group when Weyl was 3 years old and listed the orders of the Coxeter was born ."
36.
The form of the characteristic equation is related to the planes, making it possible to relate its algebraic properties like repeated roots to the bivectors, where repeated bivector magnitudes have particular geometric interpretations.
37.
While increasing " T " reduces the magnitude of the exponent in the characteristic equation, the value of " I 0 " increases exponentially with " T ".
38.
For instance, if you've got a second-order diff equation which you solve the characteristic equation for, the real roots will represent exponential solutions and the imaginary ones will represent periodic ones.
39.
If a root of the process's characteristic equation is larger than 1, then it is called an "'explosive process "', even though such processes are often inaccurately called unit roots processes.
40.
In linear algebra, the "'Cayley Hamilton theorem "'( named after the mathematicians Arthur Cayley and William Rowan Hamilton ) states that every square matrix over a commutative ring ( such as the characteristic equation.
