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- Thread starter matqkks
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lurflurf

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1)Ordinary differential equations

suppose we have some vector space of functions closed under differentiation

ie x'=Ax

if p is the characteristic polynomial

p(A)x=p(D)x=0

so we can solve for x

2)basis for F[A] (all operator polynomials)

clearly dim(F[A])<=n^2

but the Cayley Hamilton Theorem gives an improvement to

dim(F[A])<=n

thus we can reduce operator polynomials (though not in the best way possible in general)

invert nonsingular operators

reduce some infinite series to finite series

like exp(At)~I+A+(1/2)A^2+...+(1/n!)A^n+...

which we can also use to solve Ordinary differential equations

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I have come across the Cayley-Hamilton theorem in a college rheology class. Rheology is basically the study of material behaviour, and so rheologists look for equations that can describe materials.

Elastic solids are simple in this regard, and can be with a linear relationship between stress state (described by the stress tensor [tex]\mathbf{T}[/tex]) and deformation state (described by the deformation tensor [tex]\mathbf{B}[/tex]):

[tex] \mathbf{T} = c\mathbf{B}[/tex]

Some materials behave non-linearly, so you can try describing them by taking a power series in stead of just a constant function

[tex]\mathbf{T} = f_0\mathbf{B}^0 + f_1\mathbf{B} + f_2\mathbf{B}^2 + f_3\mathbf{B}^3 + ...[/tex]

Using the Cayley-Hamilton theorem, we can express all the higher power tensors in terms of the lower power tensors and the invariants of [tex]\mathbf{B}[/tex]. With some elimination you can end up with the expression:

[tex]\mathbf{T} = g_0\mathbf{I} + g_1\mathbf{B} + g_2\mathbf{B}^{-1}[/tex]

Some further analysis shows that [tex]g_0 = -p[/tex] (pressure) and [tex]g_1[/tex] and [tex]g_2[/tex] are functions of the first and second invariant of [tex]\mathbf{B}[/tex].

This has now allowed rheologists to express complex material behaviour in terms of just the deformation state tensor and its invariant.

A similar analysis can be employed for describing viscous fluid behaviour. The only big difference is that not the deformation state tensor, but the rate of deformation tensor is used in the power expansion.

[tex]\mathbf{T} = -p\mathbf{I} + \eta_1\mathbf{2D} + \eta_2(\mathbf{2D})^2[/tex]

With [tex]eta_1[/tex] and [tex]eta_2[/tex] a function of the second and third invariants of [tex]\mathbf{2D}[/tex].

However, it tuned out that this equation gave the wrong predictions. The error arose with the last term, so it was dropped and the general equation describing viscous fluids:

[tex]\mathbf{T} = -p\mathbf{I} + \eta_1\mathbf{2D}[/tex]

Hopefully this has given you some idea of the application of the Cayley-Hamiltonian theorem. :-)

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An application

Take a square matrix A of order n and a polynomial T(x) of degree r, such that r>n. How can we compute T(A)?

Of course a direct computation is always possible, but perhaps not so illuminating.

Denote by PA(x) the characteristic polynomial of A and then use Euclide's algorithm: there exists a unique ordered pair of polynomials (Q(x),R(x)) such that T(x)=Q(x) PA(x) +R(x) and .

By Cayley-Hamilton's Theorem , we have:

T(A) = R (A)

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