characteristic polynomial

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English

Noun

characteristic polynomial (plural characteristic polynomials)

  1. (linear algebra) The polynomial produced from a given square matrix by first subtracting the appropriate identity matrix multiplied by an indeterminant and then calculating the determinant.
    The characteristic polynomial of is .
    The characteristic polynomial of a matrix M is , where denotes the trace of M and denotes the determinant of M.
    The characteristic polynomial of a matrix M is , where denotes the adjugate of M.
  2. (mathematics) A polynomial P(r) corresponding to a homogeneous, linear, ordinary differential equation P(D) y = 0 where D is a differential operator (with respect to a variable t, if y is a function of t).

Usage notes

Equally many authors instead subtract the matrix from the indeterminant times the identity matrix. The result differs only by a factor of -1, which turns out to be unimportant in the theory of characteristic polynomials.

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