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In Lanczos algorithm, can we choose the staring vector $v$ to be the first eigenvector of the input matrix $A$? How can we select it? and why $v$ need to have norm 1?

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Of course you can! The starting vector is completely arbitrary. You'll get breakdown at step 1 if you do though. Be sure to check what it entails.

This will typically happen by a "happy accident" rather than by your explicit choice: if you already know an eigenvector then you don't need to run Lanczos to compute it (or you need to run it with a different starting vector so that you can compute other eigenvectors).

Why does $v$ need to have norm 1? It does need to (at least the way the method is usually defined); but typically the very first thing you do in the algorithm is normalizing it by replacing it with $\frac{1}{\|v\|} v$, because you aim to construct an orthonormal basis.

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