Answer by Gil Kalai for Applications of the Cayley-Hamilton theorem
There is a very important theorem by Procesi that derives all polynomial identities of the ring of matrices from the Cayley-Hamilton theorem. For a review of rings with polynomial identities see this...
View ArticleAnswer by Al-Amrani for Applications of the Cayley-Hamilton theorem
Resolution of (homogeneous) linear differential systems (HLDS).Unless I am wrong, nobody above mentioned the use of Cayley-Hamilton Theorem (CHT) in computing exp(A) (A a nxn complex martix). This is...
View ArticleAnswer by Ian Morris for Applications of the Cayley-Hamilton theorem
I hope that readers will forgive my answering a slightly different question by describing what I perceive to be the value of the Cayley-Hamilton Theorem in general: it captures in an essential way the...
View ArticleAnswer by Richard Zhang for Applications of the Cayley-Hamilton theorem
Here is a beautiful result from numerical analysis. Given any nonsingular $n\times n$ system of linear equations $Ax=b$, an optimal Krylov subspace method like GMRES must necessarily terminate with the...
View ArticleAnswer by Abdelmalek Abdesselam for Applications of the Cayley-Hamilton theorem
For recent examples of perhaps surprising applications in quite advanced research topics, one can note the following:A graphical proof of the Cayley-Hamilton Theorem inspired Prop 7.1 in this work of...
View ArticleAnswer by Victor Wang for Applications of the Cayley-Hamilton theorem
Cayley-Hamilton can be used to prove the classification of finite-dimensional real division algebras (though there are probably many other proofs).
View ArticleAnswer by Jairo Bochi for Applications of the Cayley-Hamilton theorem
Cayley-Hamilton theorem can be used to prove Gelfand's formula (whose usual proofs rely either on complex analysis or normal forms of matrices).Let $A$ be a $d\times d$ complex matrix, let $\rho(A)$...
View ArticleAnswer by Peter Mueller for Applications of the Cayley-Hamilton theorem
A nice application is Ilya Bogdanov's mathoverflow answer which proves in less than $3$ lines that the elements of $\text{GL}(n,\mathbb F_q)$ have order at most $q^n-1$.
View ArticleAnswer by Denis Serre for Applications of the Cayley-Hamilton theorem
The Cayley-Hamilton theorem is used in the proof of a Theorem of Jacobson:Let $k$ be a field of characteristic $0$. Let $A,B\in{\bf M}_n(k)$ be such that $[[A,B],A]=0_n$. Then $[A,B]$ is nilpotent.A...
View ArticleAnswer by darij grinberg for Applications of the Cayley-Hamilton theorem
This might be somewhat vague, but the Cayley-Hamilton theorem is important in Galois theory. Specifically, if $L/K$ is a finite extension of fields, then an element $a \in L$ has both a minimal...
View ArticleAnswer by Steve Huntsman for Applications of the Cayley-Hamilton theorem
Cayley-Hamilton is part of what makes the Wiedemann algorithm for efficient sparse linear algebra work so well
View ArticleAnswer by José Figueroa-O'Farrill for Applications of the Cayley-Hamilton...
This may be too trivial for what you have in mind, but I've always found Cayley-Hamilton useful to calculate eigenvectors of a square matrix given the eigenvalues, without having to solve any...
View ArticleAnswer by Todd Trimble for Applications of the Cayley-Hamilton theorem
Cayley-Hamilton can be useful in commutative algebra. Related to its close connection with Nakayama's lemma as mentioned in a comment by Qiaochu (see also Wikipedia), see for example the development...
View ArticleAnswer by Piyush Grover for Applications of the Cayley-Hamilton theorem
In control theory, it is used to define very important concepts of observability and controllability of linear systems. http://www.ece.rutgers.edu/~gajic/psfiles/chap5traCO.pdf
View ArticleApplications of the Cayley-Hamilton theorem
The Cayley-Hamilton theorem is usually presented in standard undergraduate courses in linear algebra as an important result. Recall that it says that any square matrix is a "root" of its own...
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