# Do either of $c\cdot I_n-A$ or $A-c\cdot I_n$ have a name?

Let $$A$$ be an $$n\times n$$ matrix and let $$c$$ be a scalar. The matrices $$c\cdot I_n-A$$ and $$A-c\cdot I_n$$ are used to say interesting things about $$A$$. Do either of $$c\cdot I_n-A$$ and $$A-c\cdot I_n$$ have a name?

It seems like calling $$A-c\cdot I_n$$ something like the "shift of $$A$$ by a factor of $$c$$" makes sense, but "shift matrices" usually refer to binary matrices whose only nonzero entries are either on the superdiagonal or subdiagonal.

I suppose it's also worth asking if either of $$c\cdot I-T$$ or $$T-c\cdot I$$ have a name if $$T:V\to V$$ is a linear endomorphism of a vector space $$V$$ and $$I:V\to V$$ is the identity.

I am not aware of any special name for an operator or matrix of the form $$\lambda I - T$$ (or $$cI - A$$) in either linear algebra or functional analysis. However, some possibilities are

• an inverse of the resolvent,
• a pseudoinverse of the resolvent, or
• a pre-resolvent.

All three of these are meant to capture the the notion that $$cI - A$$ has a relation to the resolvent operator, though the exact nature of that relation is a little delicate. More context is provided below.

Given that endomorphisms of vector spaces are mentioned, it may be that much of the following is redundant for the asker. However, I think that the following discussion helps to justify the terminology suggested above.

In functional analysis, if $$T$$ is a bounded linear operator on a Banach space (a topological vector space possessing a complete norm) and $$\lambda \in \mathbb{C}$$, then the resolvent operator (corresponding to $$T$$ at $$\lambda$$) is defined to be $$R_{\lambda} := (\lambda I - T)^{-1},$$ where $$I$$ is the identity operator.

The resolvent operator plays a special role in functional analysis: the resolvent set of $$T$$ consists of all of the values of $$\lambda$$ such that $$R_{\lambda}$$ is a bounded linear operator (e.g. it exists and is bounded), while the spectrum of $$T$$ consists of all other values of $$\lambda$$. In the case that $$T : \mathbb{R}^n \to \mathbb{R}^n$$, then $$T$$ may be represented by an $$n\times n$$ matrix, and the spectrum of $$T$$ will correspond to the set of eigenvalues. The Wikipedia article on spectral theory offers a reasonable summary.

 And now that I have checked out the asker's profile, I am reasonably certain that the context is redundant. However, it might be useful to others. ;)

• What about "the pseudoinverse of the resolvent"? The genuine inverse of $c\cdot I_n-A$ doesn't exist when $c$ is an eigenvalue, but the pseudoinverse always exists. – Brian Fitzpatrick Oct 6 at 20:10
• Or would people object to calling the pseudoinverse of $c\cdot I_n-A$ the resolvent? I'm not particularly well-acquainted with the language of spectral theory! – Brian Fitzpatrick Oct 6 at 20:36
• I like pseudoinverse. Or, perhaps, because the operator $\lambda - T$ is guaranteed to be defined and relatively well behaved (aside from potentially lacking an inverse) a pre-resolvent? Unfortunately, I, too, am not an expert in spectral theory (though, frankly, I really ought to be better versed in it...), and I am unaware of any special term for this operator in that context. – Xander Henderson Oct 6 at 23:00
• @BrianFitzpatrick I've edited the answer to include the new suggestions. – Xander Henderson Oct 6 at 23:05

It is a specific type of linear matrix pencil.