Householder QR Step

Problem

Apply one Householder QR step to the first column of matrix A (n x 2).

Build the unit Householder vector from the first column with alpha = -sign(col[0]) * ||col||_2, form H = I - 2*v*v^T, and return H*A with zeros below the first entry of column 0.

Hint

The Householder reflection maps the first column to a multiple of e1, producing a triangular first column.

Test cases

case 1

in: [[[3,4],[0,0],[0,0]]]

expected: [[-3,-4],[0,0],[0,0]]

case 2

in: [[[1,1],[1,2],[1,3]]]

expected: [[-1.7320508075688772,-3.4641016151377544],[0,0.3660254037844386],[0,1.3660254037844384]]

case 3

in: [[[1,0],[0,1],[0,0]]]

expected: [[-1,0],[0,1],[0,0]]

case 4

in: [[[2,3],[2,1],[0,0]]]

expected: [[-2.828427124746189,-2.8284271247461885],[0,-1.414213562373095],[0,0]]

Starter codePython

def householder_qr_step(A):
    # Your code here
    pass

4 test cases · run your code against them in the editor