characteristic polynomial

Noun

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

Translations

 * Estonian: karakteristlik polünoom
 * Finnish: karakteristinen polynomi
 * French: polynôme caractéristique
 * German: charakteristiches Polynom
 * Hungarian:
 * Italian: polinomio caratteristico
 * Portuguese: polinómio característico, polinômio característico
 * Romanian: polinom caracteristic
 * Spanish: polinomio característico
 * Swedish: