Let be two matrices. If don’t commute, then ; however, the two share several properties. If either or is invertible, then is conjugate to , so in particular they have the same characteristic polynomial.
What if neither nor are invertible? As it turns out, and still have the same characteristic polynomial, although they are not conjugate in general (e.g. we might have but nonzero). There are several ways of proving this result, which implies in particular that and have the same eigenvalues.
What if are linear transformations on an infinite-dimensional vector space? Do and still have the same eigenvalues in an appropriate sense? As it turns out, the answer is yes, and the key lemma in the proof is an interesting piece of “noncommutative high school algebra.”
Proposition: Let be two matrices over an algebraically closed field . Then and have the same characteristic polynomial.
(Note that this is a polynomial identity in the entries of regarded as formal variables, so to prove it identically (in particular for all commutative rings ) it suffices to prove it over a fixed algebraically closed field, e.g. we could take .)
Proof 1. This is clear if is invertible since . The invertible matrices are Zariski-dense in all matrices (when they are also dense in the usual topology), so the result follows in general.
Proof 2. Recall that . (This is straightforward to prove by computation but it also has an elegant proof using the invariant description of the trace as tensor contraction , and one can also prove it in the same way as above.) By induction we conclude that for all , so the power sum symmetric polynomials in the eigenvalues of and are identical. By the Newton-Girard identities, it follows that the elementary symmetric polynomials in the eigenvalues of and are identical.
Proof 3. We work universally. Let the entries of be formal variables in a polynomial ring . Note that we have
where is another (scalar) variable and is the identity matrix. Since is an integral domain, we can cancel from both sides to obtain
What can we say beyond matrices? It might be reasonable to guess that and still have the same eigenvalues if are linear transformations on an infinite-dimensional vector space; however, since no analogue of the characteristic polynomial, the trace, or the determinant is available in general in this setting, the proofs above don’t generalize, and in fact the result is false.
Example. Consider the differential operators and acting on ( a field of characteristic zero). The former has eigenvectors with eigenvalues while the latter has eigenvectors with eigenvalues ; thus the eigenvalues of the former are while the eigenvalues of the latter are .
and provide examples of other infinite-dimensional phenomena too: they are endomorphisms of a vector space like matrices, but one is injective without being surjective, one is surjective without being injective, and doesn’t have any eigenvectors whatsoever (even if is algebraically closed)!
The issue that occurs in the above example is the following. Let be two linear operators on a vector space . Suppose that is an eigenvector for , thus
for some . Then
so that is an eigenvector for with the same eigenvalue, unless it is equal to zero. But in that case , so . If , then this cannot occur. So we have proven the following.
Proposition: Let be two linear transformations on a vector space . Then is a nonzero eigenvalue of if and only if it is a nonzero eigenvalue of .
Thus the only possible discrepancy in the sets of eigenvalues occurs when , and the example of shows that this can occur.
Unfortunately, as we have seen, if is infinite-dimensional there are linear transformations with no eigenvectors whatsoever, such as acting on . Is there no hope of generalizing the above result to such linear transformations?
When we talk about eigenvalues, what are we really talking about? On a finite-dimensional vector space , to say that a linear transformation has an eigenvector with eigenvalue is to say that , or . This is true if and only if fails to be invertible, which suggests the following definition.
Definition: Let be a linear transformation on a vector space over a field . The spectrum consists of all such that is not invertible.
If is not invertible because it is not injective, then is an eigenvalue; in functional analysis terms, lies in the point spectrum . However, may also fail to be invertible because it is not surjective, and in this case the spectrum is strictly larger than the point spectrum.
Example. The linear operator acting on has spectrum all of even though it has no point spectrum.
Example. The linear operator acting on has empty spectrum.
Example. Let be a topological space, be the space of continuous functions , and let be multiplication by some function . Then the spectrum of is precisely the range of . (This is one basic reason it’s reasonable to think of eigenvalues of operators in quantum mechanics as being values of functions such as position and momentum.) In many cases, has no point spectrum (e.g. ).
There is a nice connection between spectra in this sense and spectra of rings in algebraic geometry. A simple version is as follows. If is a linear transformation of a finite-dimensional vector space over an algebraically closed field , then we can consider the ring generated by in . This ring is isomorphic to where is the minimal polynomial of , so the spectrum of can naturally be identified with the set of eigenvalues (that is, the spectrum) of !
A more general connection can be obtained by generalizing the definition of spectrum.
Definition: Let be a field and a -algebra. Let . The spectrum of consists of all such that is not invertible in .
(We recover the definition applied to linear operators by taking .)
Now observe that if is commutative, then is not invertible if and only if it is contained in a maximal ideal . This maximal ideal has the property that in , thus thinking of as a function on it follows that takes on every value in its spectrum at an appropriate maximal ideal! If the residue fields of are all (which occurs for example if is algebraically closed and is finitely-generated by the Nullstellensatz) then the converse is also true.
We are now ready to state the appropriate generalization of the proposition above about characteristic polynomials.
Proposition: Let be two elements of a -algebra . Then .
The crucial lemma
The proposition claims that if is nonzero, then is invertible if and only if is invertible. Equivalently, is invertible if and only if is invertible. By setting , we see that the following piece of “noncommutative high school algebra” implies and in fact generalizes our proposition.
Lemma: Let be elements of a ring . Then is invertible if and only if is invertible.
This lemma is somewhat infamous for having the following “proof.” Pretend that it makes sense to write
in a general ring. Then
And indeed, if we write , then we dutifully find that
Halmos once posed the problem of explaining in what sense the geometric series proof works. There is a discussion of this problem on MO which I would summarize as follows. The universal ring describing this problem is the free ring
on two elements , and an inverse to . If we show that has an inverse in this ring, then we are done by the universal property. And the idea is that this ring ought to embed in a suitable ring of formal power series where one can make sense of geometric series expansions. To make this easier, we’ll work with a different ring
where is a new central variable, and we want to embed into by sending to . In this ring the geometric series argument makes perfect sense and proves that is an inverse to . Applying the evaluation homomorphism out of obtained by setting then solves our original problem.
However, it does not seem trivial to me to prove that the natural map from to is actually injective, and nobody in the MO discussion above seems to actually prove this.