Overrated Characteristic Polynomial Criticisms | Generated by AI
Question: What are some unpopular or controversial opinions about the characteristic polynomial in linear algebra?
Answer:
Here are several genuinely unpopular (or at least minority) opinions about the characteristic polynomial that many mathematicians and educators quietly hold but strongly hold, even if they don’t always say them out loud in classrooms or textbooks:
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The characteristic polynomial is vastly overemphasized in introductory linear algebra courses.
For most practical computations (eigenvalues, diagonalizability checks, Jordan form, etc.), the characteristic polynomial is one of the worst possible tools: it’s numerically unstable, high-degree, and gives you almost no geometric insight. Yet almost every textbook spends chapters on it because of historical inertia. -
The determinant definition det(A − λI) is pedagogically terrible.
Most students remember “det(A − λI) = 0 gives eigenvalues,” but very few ever understand why. The trace-definition version (monic polynomial whose roots are the eigenvalues) or the “smallest polynomial p such that p(A) = 0 has degree at most n” are conceptually cleaner, but almost never introduced first. -
The characteristic polynomial is essentially useless for proving deep theorems.
Major results like Cayley–Hamilton, spectral theorem, Jordan form, Perron–Frobenius, etc., are usually proved without ever writing down the full characteristic polynomial explicitly. The polynomial itself is a blunt instrument; operator-theoretic or representation-theoretic tools are sharper. -
Calling it “characteristic” is a historical misnomer.
It’s invariant under similarity, yes, but so are dozens of other things (trace, determinant, eigenvalues with multiplicity, Jordan block structure up to permutation, etc.). The minimal polynomial is often more “characteristic” of the operator than the characteristic one. -
For random matrices or numerical linear algebra, nobody in their right mind computes the characteristic polynomial.
Real algorithms (QR, Arnoldi, Jacobi–Davidson, etc.) never form det(A − λI). The only places you still see it are in exact symbolic computation or theoretical physics papers that want a closed-form expression at all costs. -
The theorem “similar matrices have the same characteristic polynomial” is overhyped.
It’s true, but the converse is spectacularly false, and students are often left with the subconscious impression that two matrices with the same characteristic polynomial are “almost the same,” when they can have completely different Jordan forms. -
In infinite dimensions, the concept basically collapses, so fixating on it early gives students bad intuition for functional analysis later.
In short, many working mathematicians view the characteristic polynomial as a 19th-century artifact that refuses to die because it’s easy to define and gives a quick way to introduce eigenvalues, not because it’s actually good for anything serious.
(You’ll still find it in every textbook because “that’s how we all learned it,” but ask around privately at conferences and you’ll hear a lot of eye-rolling.)