Characteristic polynomial information

In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues...

of a polynomial with degree 5 or more. (Generality matters because any polynomial with degree n {\displaystyle n} is the characteristic polynomial of some...

obtained by equating to zero the characteristic polynomial of a matrix or of a linear mapping Method of characteristics, a technique for solving partial...

isomorphic matroids have the same polynomial. The characteristic polynomial of M – sometimes called the chromatic polynomial, although it does not count colorings...

meanings in specific domains Characteristic polynomial, a polynomial associated with a square matrix in linear algebra Characteristic subgroup, a subgroup that...

differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the...

computationally much more efficient. Determinants are used for defining the characteristic polynomial of a square matrix, whose roots are the eigenvalues. In geometry...

homogeneous, the coefficients determine the characteristic polynomial (also "auxiliary polynomial" or "companion polynomial") p ( λ ) = λ n − a 1 λ n − 1 − a 2...

all eigenvalues of the matrix lie in K, or equivalently if the characteristic polynomial of the operator splits into linear factors over K. This condition...

coefficients of p in reverse order. Reciprocal polynomials arise naturally in linear algebra as the characteristic polynomial of the inverse of a matrix. In the special...

the characteristic polynomial of A. So the algebraic multiplicity is the multiplicity of the eigenvalue as a zero of the characteristic polynomial. Since...

Bernstein polynomial Characteristic polynomial Minimal polynomial Invariant polynomial Abel polynomials Actuarial polynomials Additive polynomials All one...

A} ; this polynomial is the minimal polynomial. Any polynomial which annihilates A {\displaystyle A} (such as the characteristic polynomial) is a multiple...

)}{\det \mathbf {A} }},} where xi is the ith entry of x. Let the characteristic polynomial of A be p ( s ) = det ( s I − A ) = ∑ i = 0 n p i s i ∈ R [ s...

elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed...

vectors. The characteristic function of a cooperative game in game theory. The characteristic polynomial in linear algebra. The characteristic state function...

{1}{\phi (B)}}\varepsilon _{t}\,.} When the polynomial division on the right side is carried out, the polynomial in the backshift operator applied to ε t...

of p ( x ) {\displaystyle p(x)} , while the characteristic polynomial as well as the minimal polynomial of C ( p ) {\displaystyle C(p)} are equal to...

F, then the polynomial (x − a1)(x − a2) ⋯ (x − an) + 1 has no zero in F. However, the union of all finite fields of a fixed characteristic p is an algebraically...

of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more...

First, it requires finding all eigenvalues, say as roots of the characteristic polynomial, but it may not be possible to give an explicit expression for...

the ring of polynomials, of the matrix (with polynomial entries) XIn − A (the same one whose determinant defines the characteristic polynomial). Note that...

{\displaystyle u=e^{\lambda t}} . That substitution yields the characteristic polynomial p L ( λ ) = λ n + a 1 λ n − 1 + ⋯ + a n − 1 λ + a n {\displaystyle...

In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the...