Search results
Results from the WOW.Com Content Network
Urbain Le Verrier (1811–1877) The discoverer of Neptune.. In mathematics (linear algebra), the Faddeev–LeVerrier algorithm is a recursive method to calculate the coefficients of the characteristic polynomial = of a square matrix, A, named after Dmitry Konstantinovich Faddeev and Urbain Le Verrier.
The roots of the characteristic polynomial () are the eigenvalues of ().If there are n distinct eigenvalues , …,, then () is diagonalizable as () =, where D is the diagonal matrix and V is the Vandermonde matrix corresponding to the λ 's: = [], = [].
This method is very easy to implement using dynamic arrays on a computer. It also tells whether all the modulus of the roots (complex and real) lie inside the unit disc. The vector v contains the real coefficients of the original polynomial in the order from highest degree to lowest degree.
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
A polynomial satisfying the Routh–Hurwitz criterion is called a Hurwitz polynomial. 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 ( bounded ).
Such a set of matrices is more easily understood by considering the algebra of matrices it generates, namely all polynomials in the , denoted [, …,]. Simultaneous triangularizability means that this algebra is conjugate into the Lie subalgebra of upper triangular matrices, and is equivalent to this algebra being a Lie subalgebra of a Borel ...
For a first-order PDE, the method of characteristics discovers so called characteristic curves along which the PDE becomes an ODE. [ 1 ] [ 2 ] Once the ODE is found, it can be solved along the characteristic curves and transformed into a solution for the original PDE.