characteristic polynomial

English

Noun

characteristic polynomial (plural characteristic polynomials)

(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 ${\\displaystyle \\textstyle \\left({\\begin{array}{cc}1&4\\\\3&-5\\end{array}}\\right)}$ is ${\\displaystyle \\textstyle \\left\\vert {\\begin{array}{cc}1-x&4\\\\3&-5-x\\end{array}}\\right\\vert =x^{2}+4x-17}$ .
The characteristic polynomial of a ${\\displaystyle 2\\times 2}$ matrix M is ${\\displaystyle \\lambda ^{2}-{\\mbox{tr}}(M)\\lambda +{\\mbox{det}}(M)}$ , where ${\\displaystyle {\\mbox{tr}}(M)}$ denotes the trace of M and ${\\displaystyle {\\mbox{det}}(M)}$ denotes the determinant of M.
(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.

Related terms

characteristic equation
characteristic root
characteristic vector

Translations

linear algebra

Estonian: karakteristlik polünoom
Finnish: karakteristinen polynomi
French: polynôme caractéristique m
German: charakteristiches Polynom n
Hungarian: karakterisztikus polinom

Italian: polinomio caratteristico m
Portuguese: polinómio característico m, (Brazil) polinômio característico m
Romanian: polinom caracteristic n
Spanish: polinomio característico m
Swedish: karakteristiskt polynom (sv) n