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...

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

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

is a (polynomial) multiple of μA. The following three statements are equivalent: λ is a root of μA, λ is a root of the characteristic polynomial χA of...

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

complex numbers or the integers) satisfies its own characteristic equation. The characteristic polynomial of an n × n matrix A is defined as p A ( λ ) = det...

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...

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...

of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomial P in one variable...

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...

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

t^{n-1}} in the characteristic polynomial, possibly changed of sign, according to the convention in the definition of the characteristic polynomial. If A is...

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

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

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

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

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

)}{\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...

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...

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...

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...

{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...

reciprocal characteristic polynomial. For example, if the taps are at the 16th, 14th, 13th and 11th bits (as shown), the feedback polynomial is x 16 +...

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...

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

{\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...

Note that all characteristic polynomials and minimal polynomials of A are annihilating polynomials. In fact, every annihilating polynomial is the multiple...

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

Routh proposed in 1876 to determine whether all the roots of the characteristic polynomial of a linear system have negative real parts. German mathematician...