The resolvent matrix of a matrix $A$ is defined as$$ R_{A}(s)=(sI-A)^{-1}.$$

Note: $I$ is the identity matrix and $s$ is a complex variable. Also note that $R_{A}(s)$ is undefined on $Sp(A)$ (the spectrum of $A$ ).

More generally, let $A$ be a unital algebra over the field of complex numbers $\mathbb{C}$ . The resolvent $R_x$ of an element $x\in A$ is a function from $\mathbb{C}-Sp(x)$ to $A$ given by$$ R_x(s)=(s\cdot 1-x)^{-1}$$ where $Sp(x)$ is the spectrum of $x$ : $Sp(x)=\lbrace t\in \mathbb{C}\mid t\cdot 1 -x\mbox{ is not invertible in }A\rbrace$ .

If $A$ is commutative and $s\notin Sp(x)\cup Sp(y)$ , then $R_x(s)-R_y(s)=R_x(s)R_y(s)(x-y)$ .