Characteristic polynomial

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The characteristic polynomial of a linear operator refers to the polynomial whose roots are the eigenvalues of the operator.

Definition

Suppose $A$ is a $n \times n$ matrix (over a field $K$). Then the characteristic polynomial of $A$ is defined as $P_A(t) = \Det(tI - A)$ (Error compiling LaTeX. ! Undefined control sequence.), which is a $n$th degree polynomial in $t$.

Written out,

\[P_A(t) = \begin{vmatrix}t-a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & t-a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & t-a_{nn}\end{vmatrix}\]

Properties

An eigenvector $\bold{v} \in K^n$ is a non-zero vector that satisfies the relation $A\bold{v} = \lambda\bold{v}$, for some scalar $\lambda \in K$. In other words, applying a linear operator to an eigenvector causes the eigenvector to dilate. The associated number $\lambda$ is called the eigenvalue.

There are at most $n$ 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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