Characteristic polynomial

English

Noun

1. Template:Lb 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 $\left(\begin{array}{cc}1& 4\\ 3& -5\end{array}\right)$ is $\left|\begin{array}{cc}1-x& 4\\ 3& -5-x\end{array}\right|=x^2+4x-17$.

   The characteristic polynomial of a $2\times 2$ matrix M is ${\lambda }^2-\text{tr}(M)\lambda +\text{det}(M)$, where $\text{tr}(M)$ denotes the trace of M and $\text{det}(M)$ denotes the determinant of M.

   The characteristic polynomial of a $3\times 3$ matrix M is $-{\lambda }^3+\text{tr}(M){\lambda }^2-\text{tr}(\text{adj}(M))\lambda +\text{det}(M)$, where $\text{adj}(M)$ denotes the adjugate of M.

2. Template:Lb 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

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See also

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