characteristic polynomial

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characteristic polynomial

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

   The characteristic polynomial of a ${\displaystyle 3\times 3}$ matrix M is ${\displaystyle -\lambda ^{3}+{\mbox{tr}}(M)\lambda ^{2}-{\mbox{tr}}({\mbox{adj}}(M))\lambda +{\mbox{det}}(M)}$, where ${\displaystyle {\mbox{adj}}(M)}$ 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

See also

• minimal polynomial
• eigenvector
• eigenvalue

Further reading

• Cayley–Hamilton theorem on Wikipedia.Wikipedia