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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2020

Paper 2

The $m$th <Krylov subspace> is
$$
\boxed{
\mathcal K_m(A,v)
=\operatorname{span}\{v,Av,\ldots,A^{m-1}v\}
}.
$$
Because $A$ has a complete eigenbasis, decompose
$$
v=v_1+\cdots+v_s,
$$
where $v_j$ is the projection of $v$ onto the eigenspace belonging to the $j$th distinct eigenvalue $\lambda_j$; some $v_j$ may vanish. Then
$$
A^kv=\sum_{j=1}^s\lambda_j^kv_j.
$$
Every generator of $\mathcal K_m(A,v)$ therefore belongs to $\operatorname{span}\{v_1,\ldots,v_s\}$, and
$$
\boxed{\dim\mathcal K_m(A,v)\leq s\quad\text{for every }m}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 40E / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 40E / a / Solution

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Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 40E / a / Solution

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