characteristic polynomial

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 $\textstyle \left({\begin{array}{cc}1&4\\3&-5\end{array}}\right)$ 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.

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.

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 (hu)
• 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

See also

• minimal polynomial
• eigenvector
• eigenvalue

Further reading

• Cayley–Hamilton theorem on Wikipedia.Wikipedia