september 9

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\def\Field{\mathbb{F}}
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Linear operators; normal forms

Linear operators between different spaces are easy to normalize: the rank is the only invariant; the classification is discrete.

For endomorphisms, the situation is more involved.

Diagonalization corresponds to splitting the space into the simplest pieces, 1-dimensional. It is not always possible.

Example: differentiation in the basis \(x^k/k!\).

In this case – apply Jordan normal form

Jordan blocks (example: differentiating quasi polynomials). Structure theorem.

Other normal forms: cyclic vectors…

Other classification problems (quivers…)

Cayley-Hamilton Theorem

Powers of an \(n\times n\) matrix sit in \(n^2\)-dimensional space; there is a lot of room – but \(n+1\) of them are already linearly dependent!

Also helps find functions of matrices, such as

matrix exponential

which is instrumental in finding solutions to linear systems of differential equations.

Computing the matrix exponentials.

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