In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues...
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Eigenvalues and eigenvectors (redirect from Characteristic value)
of a polynomial with degree 5 or more. (Generality matters because any polynomial with degree n {\displaystyle n} is the characteristic polynomial of some...
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Matroid (redirect from Characteristic polynomial of matroids)
isomorphic matroids have the same polynomial. The characteristic polynomial of M – sometimes called the chromatic polynomial, although it does not count colorings...
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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...
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meanings in specific domains Characteristic polynomial, a polynomial associated with a square matrix in linear algebra Characteristic subgroup, a subgroup that...
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obtained by equating to zero the characteristic polynomial of a matrix or of a linear mapping Method of characteristics, a technique for solving partial...
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Jordan normal form (section Characteristic polynomial)
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...
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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...
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Bernstein polynomial Characteristic polynomial Minimal polynomial Invariant polynomial Abel polynomials Actuarial polynomials Additive polynomials All one...
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Newton's identities (redirect from Newton's theorem on symmetric polynomials)
of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomial P in one variable...
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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...
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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...
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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...
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Adjugate matrix (section Characteristic polynomial)
)}{\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...
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especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally...
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vectors. The characteristic function of a cooperative game in game theory. The characteristic polynomial in linear algebra. The characteristic state function...
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Linear recurrence with constant coefficients (redirect from Characteristic equation (of difference equation))
homogeneous, the coefficients determine the characteristic polynomial (also "auxiliary polynomial" or "companion polynomial") p ( λ ) = λ n − a 1 λ n − 1 − a 2...
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Determinant (category Homogeneous polynomials)
computationally much more efficient. Determinants are used for defining the characteristic polynomial of a square matrix, whose roots are the eigenvalues. In geometry...
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elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed...
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the characteristic polynomial of A. So the algebraic multiplicity is the multiplicity of the eigenvalue as a zero of the characteristic polynomial. Since...
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A} ; this polynomial is the minimal polynomial. Any polynomial which annihilates A {\displaystyle A} (such as the characteristic polynomial) is a multiple...
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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...
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Transfer function (redirect from Transfer characteristic)
{\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...
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{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...
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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...
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Algebraically closed field (redirect from Relatively prime polynomials)
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...
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graph. Its characteristic polynomial is − x ( x 2 − x − 3 ) ( x 2 + x − 1 ) {\displaystyle -x(x^{2}-x-3)(x^{2}+x-1)} . Its Tutte polynomial is x 4 + x...
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Routh–Hurwitz stability criterion (category Polynomials)
Routh proposed in 1876 to determine whether all the roots of the characteristic polynomial of a linear system have negative real parts. German mathematician...
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Linear-feedback shift register (redirect from Polynomial counter)
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 +...
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Note that all characteristic polynomials and minimal polynomials of A are annihilating polynomials. In fact, every annihilating polynomial is the multiple...
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