# Givens rotation vs 2x2 Householder reflection

The usual story of Givens rotations vs Householder reflections is that Householder reflections are better if you want to map a long vector to $$e_1$$, while Givens is better if you want to map a 2-vector to $$e_1$$. However, I can't find a reference which explains why Givens is better in the 2-vector case. Can anyone explain?

Typically you don't apply these matrices to a single vector; you apply them to a $$m\times n$$ matrix, where $$n$$ is large, so you can amortize the cost of computing the various coefficients. Applying a Householder reflector to a vector (with the formula $$x \mapsto x - vv^Tx$$) costs $$4m-1$$. Applying a Givens rotation works only for $$m=2$$, and costs $$6$$ operations, which is cheaper.
EDIT: To apply the Householder transformation, you could instead form the $$2\times 2$$ matrix $$Q = I-vv^T$$ and apply that to a vector, as suggested by @gTcV in the comments, for 6 operations. But it's not complicated to see that at this point $$Q$$ is essentially a Givens rotations with two signs swapped, so that's not really anything different from Givens transformations.
• @gTcV Good point -- I was referring to Householder reflectors when applied in the usual form $I-vv^T$; if you convert them to a $2\times 2$ matrix and apply them directly then the cost becomes the same. – Federico Poloni Aug 21 '19 at 14:13