Characteristic Polynomial

Definition: Characteristic Polynomial

The characteristic polynomial $p_A(x)$ of a square matrix $A\in F^{n\times n}$ the univariate polynomial obtained through the determinant and the identity matrix $I_n$ in the following way:

$$p_A(x)\stackrel{\text{def}}{=}\det (A-x{I}_n)$$

Theorem: Degree of the Characteristic Polynomial

The degree of the characteristic polynomial of a square matrix $A\in F^{n\times n}$ is $n$.

PROOF

TODO

Theorem: Finding Eigenvalues

The eigenvalues of a square matrix $A\in F^{n\times n}$ are precisely the roots of its characteristic polynomial.

$$\det (A-x{I}_n)=0$$

PROOF

TODO

Definition: Algebraic Multiplicity

If the characteristic polynomial of $A$ has a linear factorisation

$$p_A(x)=(x-{\lambda }_1{)}^{k_1}\cdots (x-{\lambda }_r{)}^{k_r}$$

over the field $F$, where ${\lambda }_1,\cdots ,{\lambda }_r(r\le n)$ are the distinct eigenvalues of $A$, then we call $k_i$ the algebraic multiplicity of ${\lambda }_i$.

NOTATION

$$\mathrm{alg}({\lambda }_i)$$