Linear and multilinear algebra; matrix theory Definition

Annihilating polynomial

Definition

Let $F$ be a field, let $V$ be a vector space over $F$ , and let $T \colon V \to V$ be an endomorphism. A polynomial $p \in F[x]$ annihilates $T$ when $p(T) = 0$ , where $p(T)$ is the value of $p$ at $T$ . Such a polynomial is an annihilating polynomial of $T$ .

Remarks

The annihilating polynomials of $T$ form an ideal of $F[x]$ . If $\dim V = n$ is finite, then a nonzero annihilating polynomial exists, because the $n^2 + 1$ maps $\id_V, T, \dots, T^{n^2}$ are linearly dependent. The monic generator of this ideal is the minimal polynomial of $T$ . For a projection $P$ , the polynomial $x^2 - x$ annihilates $P$ .

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Vector space definition
Linear map definition
Endomorphism definition
Polynomial of an endomorphism definition
Annihilating polynomial definition

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Tag	`052M4SX57M`
URI	`m2://052M4SX57M`
Kind	definition
Writing	W0 stub
MSC	15A18
Defines	Annihilating polynomial
Uses	Polynomial of an endomorphism

Annihilating polynomial

Definition

Remarks

Reading path

Where this leads

Further reading

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