• In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues...
    19 KB (3,050 words) - 23:34, 28 July 2025
  • of a polynomial with degree 5 or more. (Generality matters because any polynomial with degree n {\displaystyle n} is the characteristic polynomial of some...
    103 KB (13,663 words) - 21:59, 27 July 2025
  • isomorphic matroids have the same polynomial. The characteristic polynomial of M – sometimes called the chromatic polynomial, although it does not count colorings...
    60 KB (8,808 words) - 20:24, 29 July 2025
  • 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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  • Thumbnail for Cayley–Hamilton theorem
    complex numbers or the integers) satisfies its own characteristic equation. The characteristic polynomial of an n × n {\displaystyle n\times n} matrix A is...
    65 KB (11,376 words) - 09:35, 3 August 2025
  • homogeneous, the coefficients determine the characteristic polynomial (also "auxiliary polynomial" or "companion polynomial") p ( λ ) = λ n − a 1 λ n − 1 − a 2...
    25 KB (4,667 words) - 13:18, 19 October 2024
  • 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...
    13 KB (1,639 words) - 19:08, 30 July 2025
  • of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomial P in one variable...
    35 KB (7,650 words) - 23:11, 16 April 2025
  • 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...
    91 KB (14,412 words) - 00:41, 30 July 2025
  • 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...
    45 KB (7,479 words) - 09:50, 18 June 2025
  • {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...
    38 KB (5,837 words) - 19:46, 1 August 2025
  • Bernstein polynomial Characteristic polynomial Minimal polynomial Invariant polynomial Abel polynomials Actuarial polynomials Additive polynomials All one...
    5 KB (441 words) - 01:35, 1 December 2023
  • 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...
    37 KB (5,564 words) - 18:57, 30 July 2025
  • 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...
    30 KB (4,754 words) - 18:32, 3 July 2025
  • 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 +...
    38 KB (4,725 words) - 03:54, 18 July 2025
  • )}{\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...
    29 KB (4,813 words) - 02:50, 10 May 2025
  • the characteristic polynomial of A. So the algebraic multiplicity is the multiplicity of the eigenvalue as a zero of the characteristic polynomial. Since...
    40 KB (4,870 words) - 04:25, 26 May 2025
  • 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...
    16 KB (2,834 words) - 02:55, 22 April 2025
  • 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...
    54 KB (8,657 words) - 19:01, 29 July 2025
  • obtained by equating to zero the characteristic polynomial of a matrix or of a linear mapping Method of characteristics, a technique for solving partial...
    420 bytes (78 words) - 10:47, 30 April 2024
  • 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...
    58 KB (8,225 words) - 12:38, 18 July 2025
  • 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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  • Thumbnail for Transpose
    invariant factors, which implies they share the same minimal polynomial, characteristic polynomial, and eigenvalues, among other properties. A proof of this...
    19 KB (2,422 words) - 08:49, 10 July 2025
  • 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...
    17 KB (3,154 words) - 15:15, 30 June 2025
  • Thumbnail for Faddeev–LeVerrier algorithm
    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...
    12 KB (2,492 words) - 23:47, 28 July 2025
  • n} distinct eigenvalues in F {\displaystyle F} , i.e. if its characteristic polynomial has n {\displaystyle n} distinct roots in F {\displaystyle F}...
    27 KB (4,692 words) - 21:03, 14 April 2025
  • 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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  • on some Bi is disconnected. The characteristic polynomial of this matroid is obtained from the chromatic polynomial χ Γ ( t ) {\displaystyle \chi _{\Gamma...
    8 KB (1,185 words) - 18:52, 4 July 2025
  • elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed...
    19 KB (2,911 words) - 19:07, 30 July 2025
  • Thumbnail for Linear algebra
    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