Linear and multilinear algebra; matrix theory Lemma

Solutions of a nonhomogeneous polynomial operator equation

Statement

Let $V$ be a vector space of functions (for example, of sequences), and let $A$ be a linear operator on $V$ . Let $p(A) = c_0 A^k + c_1 A^{k-1} + \dots + c_k$ be a polynomial in $A$ with $c_0 \ne 0$ and $c_k \ne 0$ . Let $g \in V$ , and let $W$ be the set of all $f \in V$ with $p(A) f = 0$ . Suppose $f_0 \in V$ satisfies $p(A) f_0 = g$ . Then every $f \in V$ with $p(A) f = g$ can be written as $f = f_0 + f_1$ with $f_1 \in W$ .

Remarks

Proof: the operator $p(A)$ is linear, so $p(A)(f - f_0) = g - g = 0$ . Put $f_1 = f - f_0$ . The set $W$ is the kernel of $p(A)$ , and so a subspace of $V$ . The conditions $c_0 \ne 0$ and $c_k \ne 0$ are not used in this argument. Example: for $u_{n+1} - 2u_n = 1$ , the constant $-1$ is one solution, and $W$ is spanned by $2^n$ .

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Kernel of a linear map concept
Solutions of a nonhomogeneous polynomial operator equation statement

Tag	`EBR72JABBP`
URI	`m2://EBR72JABBP`
Kind	statement, lemma
Writing	W0 stub
MSC	15A06, 39A06
Status	This copy of the site has no build data. Run `m2 status EBR72JABBP`.
Uses	Kernel of a linear map
Changed	4 November 2026

Solutions of a nonhomogeneous polynomial operator equation

Statement

Remarks

Reading path

Further reading

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