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In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients.
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: = [], = [].
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
Important graph polynomials include: The characteristic polynomial, based on the graph's adjacency matrix. The chromatic polynomial, a polynomial whose values at integer arguments give the number of colorings of the graph with that many colors. The dichromatic polynomial, a 2-variable generalization of the chromatic polynomial
The characteristic polynomial of a matrix A is a scalar-valued polynomial, defined by () = (). The Cayley–Hamilton theorem states that if this polynomial is viewed as a matrix polynomial and evaluated at the matrix A {\displaystyle A} itself, the result is the zero matrix: p A ( A ) = 0 {\displaystyle p_{A}(A)=0} .
[3] [4] The characteristic equation can only be formed when the differential or difference equation is linear and homogeneous, and has constant coefficients. [1] Such a differential equation, with y as the dependent variable , superscript ( n ) denoting n th - derivative , and a n , a n − 1 , ..., a 1 , a 0 as constants,
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.