1.NounThe 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
(
1
4
3
−
5
)
{\displaystyle \textstyle \left({\begin{array}{cc}1&4\\3&-5\end{array}}\right)}
is
|
1
−
x
4
3
−
5
−
x
|
=
x
2
+
4
x
−
17
{\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
2
×
2
{\displaystyle 2\times 2}
matrix M is
λ
2
−
tr
(
M
)
λ
+
det
(
M
)
{\displaystyle \lambda ^{2}-{\mbox{tr}}(M)\lambda +{\mbox{det}}(M)}
, where
tr
(
M
)
{\displaystyle {\mbox{tr}}(M)}
denotes the trace of M and
det
(
M
)
{\displaystyle {\mbox{det}}(M)}
denotes the determinant of M.; The characteristic polynomial of a
3
×
3
{\displaystyle 3\times 3}
matrix M is
−
λ
3
+
tr
(
M
)
λ
2
−
tr
(
adj
(
M
)
)
λ
+
det
(
M
)
{\displaystyle -\lambda ^{3}+{\mbox{tr}}(M)\lambda ^{2}-{\mbox{tr}}({\mbox{adj}}(M))\lambda +{\mbox{det}}(M)}
, where
adj
(
M
)
{\displaystyle {\mbox{adj}}(M)}
denotes the adjugate of M."
2.NounA 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).