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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minimal and characteristic polynomials need not factor according to their roots (in F) alone, in other words they may have irreducible polynomial factors...
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complex numbers or the integers) satisfies its own characteristic equation. The characteristic polynomial of an n × n {\displaystyle n\times n} matrix A is...
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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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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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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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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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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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{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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Bernstein polynomial Characteristic polynomial Minimal polynomial Invariant polynomial Abel polynomials Actuarial polynomials Additive polynomials All one...
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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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may remark that if α is a root of the characteristic polynomial of multiplicity m, the characteristic polynomial may be factored as P(t)(t − α)m. Thus...
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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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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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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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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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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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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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Perron–Frobenius eigenvalue is simple: r is a simple root of the characteristic polynomial of A. Consequently, the eigenspace associated to r is one-dimensional...
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polynomial q(x) which has roots if and only if p(x) has roots. But if q(x) = xn + an − 1 xn − 1 + ⋯ + a0, then q(x) is the characteristic polynomial of...
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invariant factors, which implies they share the same minimal polynomial, characteristic polynomial, and eigenvalues, among other properties. A proof of this...
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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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Faddeev–LeVerrier algorithm (category Polynomials)
algorithm is a recursive method to calculate the coefficients of the characteristic polynomial p A ( λ ) = det ( λ I n − A ) {\displaystyle p_{A}(\lambda )=\det(\lambda...
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n} distinct eigenvalues in F {\displaystyle F} , i.e. if its characteristic polynomial has n {\displaystyle n} distinct roots in F {\displaystyle F}...
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the ring of polynomials, of the matrix (with polynomial entries) XIn − A (the same one whose determinant defines the characteristic polynomial). Note that...
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Dowling geometry (section Characteristic polynomial)
on some Bi is disconnected. The characteristic polynomial of this matroid is obtained from the chromatic polynomial χ Γ ( t ) {\displaystyle \chi _{\Gamma...
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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 polynomial det ( x I − M ) . {\displaystyle \det(xI-M).} If V is of dimension n, this is a monic polynomial of degree n, called the characteristic polynomial...
67 KB (7,974 words) - 07:18, 21 July 2025