Linear and multilinear algebra; matrix theory Definition

Characteristic polynomial

Definition

Let $K$ be a commutative ring, let $n \ge 1$ , and let $A$ be an $n \times n$ matrix over $K$ . The characteristic polynomial of $A$ is the polynomial $p_A(t) = \det(t I_n - A) \in K[t],$ where $\det$ is the determinant of a matrix with entries in the polynomial ring $K[t]$ .

Remarks

The polynomial $p_A$ is monic of degree $n$ . For $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ , $p_A(t) = t^2 - (a + d) t + (ad - bc)$ . Some authors use $\det(A - t I_n)$ , which equals $(-1)^n p_A(t)$ . Over a field, the roots of $p_A$ are the eigenvalues of $A$ .

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Identity matrix object
Matrix definition
Vector space definition
Square matrix definition
Linear map definition
Multilinear map definition
Alternating multilinear map definition
Determinant definition
Characteristic polynomial definition

Tag	`039HF58KQ5`
URI	`m2://039HF58KQ5`
Kind	definition
Writing	W1 stated
MSC	15A15, 15A18
Defines	Characteristic polynomial
Uses	Determinant, Square matrix

Characteristic polynomial

Definition

Remarks

Reading path

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