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.[1] 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.

[1] 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. ;)

  • $\begingroup$ 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. $\endgroup$ – Brian Fitzpatrick Oct 6 at 20:10
  • $\begingroup$ 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! $\endgroup$ – Brian Fitzpatrick Oct 6 at 20:36
  • $\begingroup$ 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. $\endgroup$ – Xander Henderson Oct 6 at 23:00
  • $\begingroup$ @BrianFitzpatrick I've edited the answer to include the new suggestions. $\endgroup$ – Xander Henderson Oct 6 at 23:05

It is a specific type of linear matrix pencil.


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