QR with HouseHolder I've learn that we can do the QR factorization with Householder but I wonder why do we do it from left to right and no the opposite.
Any intuition or explanation?
 A: The idea of Householder reflectors to obtain a QR factorization, is to compute  $Q_1A, Q_2Q_1A, Q_3Q_2Q_1A,\dots$ such that the first $k$ columns of $Q_k\dots Q_1 A$ are $0$ below the diagonal.
By starting from the left we ensure that multiplying $Q_2$ with $Q_1 A$ preserves that the entries of the first column stay $0$. This holds since the principal $1 \times 1$ submatrix of $Q_2$ is the identity matrix. By induction this holds for all steps.
Now assume that you want to start from the right (I assume you first want to find a rotation such that the last entry in the $n-1$th column is 0.). Let this rotation be given by $\tilde Q_1 A$. In the next step you search a rotation matrix, that sets the last two entries in the $i-2$th column to zero. Let $Q_2$ be such a matrix. This matrix can not have a identity block in the bottom right, since these entries are relevant for the setting of the zeros in the $i-2$th column. So, $Q_2Q_1 A$ now has zeros in the last two entries of the $n-2$th column, but it no longer has a zero in the last entry of the $i-1$th column, i.e. you do not get the desired QR decomposition.
The same also applies to when you use givens rotations.
