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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. [4]
We call p(λ) the characteristic polynomial, and the equation, called the characteristic equation, is an N th-order polynomial equation in the unknown λ. This equation will have N λ distinct solutions, where 1 ≤ N λ ≤ N. The set of solutions, that is, the eigenvalues, is called the spectrum of A. [1] [2] [3]
Determinants are used for defining the characteristic polynomial of a square matrix, whose roots are the eigenvalues. In geometry , the signed n -dimensional volume of a n -dimensional parallelepiped is expressed by a determinant, and the determinant of a linear endomorphism determines how the orientation and the n -dimensional volume are ...
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,
A matrix polynomial identity is a matrix polynomial equation which holds for all matrices A in a specified matrix ring M n (R). Matrix polynomials are often demonstrated in undergraduate linear algebra classes due to their relevance in showcasing properties of linear transformations represented as matrices, most notably the Cayley–Hamilton ...
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: = [], = [].
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.
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: