Characteristic polynomial
The characteristic polynomial of a linear operator refers to the polynomial whose roots are the eigenvalues of the operator.
Definition
Suppose is a
matrix (over a field
). Then the characteristic polynomial of
is defined as $P_A(t) = \Det(tI - A)$ (Error compiling LaTeX. Unknown error_msg), which is a
th degree polynomial in
.
Written out,
![\[P_A(t) = \begin{vmatrix}t-a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & t-a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & t-a_{nn}\end{vmatrix}\]](http://latex.artofproblemsolving.com/4/b/8/4b8ae89c0afd02be56b89a995730527b3392a518.png)
Properties
An eigenvector is a non-zero vector that satisfies the relation
, for some scalar
. In other words, applying a linear operator to an eigenvector causes the eigenvector to dilate. The associated number
is called the eigenvalue.
There are at most distinct eigenvalues, whose values are exactly the roots of the characteristic polynomial of the square matrix.
By the Hamilton-Cayley Theorem, the character polynomial of a square matrix applied to the square matrix itself is zero.
Linear recurrences
Problems
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