3
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

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. ;)

$\endgroup$
  • $\begingroup$ 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
2
$\begingroup$

It is a specific type of linear matrix pencil.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.