Cayley-Hamilton theorem

Thus one can see that the Cayley-Hamilton theorem must be true.

This follows from the previous property by the Cayley-Hamilton theorem.

The eigenvalues are the roots of the Cayley-Hamilton theorem.

Eigenvectors can be found by exploiting the Cayley-Hamilton theorem.

The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation.

There is a great variety of such proofs of the Cayley-Hamilton theorem, of which several will be given here.

One of the proofs for Cayley-Hamilton theorem above bears some similarity to the argument that .

This assertion is precisely a generalized version of the Cayley-Hamilton theorem, and the proof proceeds along the same lines.

Cayley-Hamilton theorem.

First, in Cayley-Hamilton theorem, p(A) is an nxn matrix.

The main ingredients for the following proof are the Cayley-Hamilton theorem and the fundamental theorem of algebra.

Note that by using the Cayley-Hamilton theorem and Vandermonde-type matrices, a solution may be given in a simple form.

This proof uses just the kind of objects needed to formulate the Cayley-Hamilton theorem: matrices with polynomials as entries.

Several results in linear algebra, such as Cramer's Rule and the Cayley-Hamilton theorem, have simple diagrammatic proofs.

The usual proof of this uses the following variant of the Cayley-Hamilton theorem on determinants (or simply Cramer's rule.)

In view of the Cayley-Hamilton theorem, A satisfies its own characteristic equation; i.e. Postmultiply this by any arbitrary column co and write.

Note also that the result of this computation is the scalar 0, while the Cayley-Hamilton theorem says it should be a 'n'x'n' matrix with all entries zero.

The minimal polynomial always divides the characteristic polynomial, which is one way of formulating the Cayley-Hamilton theorem (for the case of matrices over a field).

When the ring is a field, the Cayley-Hamilton theorem is equivalent to the statement that the minimal polynomial of a square matrix divides its characteristic polynomial.

In the commutative case, the lemma is a simple consequence of a generalized form of the Cayley-Hamilton theorem, an observation made by Michael Atiyah (1969).

By the Cayley-Hamilton theorem, every matrix satisfies its characteristic polynomial, and a simple transformation allows to find the matrix inverse in NC.

In particular there is natural definition of the determinant for them and most linear algebra theorems like Cramer's rule, Cayley-Hamilton theorem, etc. hold true for them.

The entities may be matrices; for instance the Cayley-Hamilton theorem applied to a matrix A equates a certain polynomial expression in A to the null matrix.

None of these computations can show however why the Cayley-Hamilton theorem should be valid for matrices of all possible sizes n, so a uniform proof for all n is needed.

He was also the first to introduce the notion of rational approximations of functions (nowadays known as Padé approximants), and gave the first full proof for the Cayley-Hamilton theorem.