Gradient of the Spectral Norm

Problem

For a matrix A with a unique largest singular value, the gradient of the spectral norm ||A||_2 is u v^T, where u and v are the top left and right singular vectors.

Given the four entries of a 2x2 matrix, compute the top singular pair from the eigen-decomposition of A^T A and return u v^T as a nested list. When the top singular value is tied, use the first coordinate direction.

Hint

The top right singular vector is the top eigenvector of A^T A, and u = A v / ||A v||.

Test cases

case 1

in: [3,0,0,1]

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

case 2

in: [0,1,0,0]

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

case 3

in: [2,0,0,2]

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

case 4

in: [1,2,3,4]

expected: [[0.2330424601316968,0.3306883952871801],[0.526804530425364,0.7475382155592233]]

Starter codePython

def spectral_norm_gradient(a, b, c, d):
    # Return u @ v^T
    # Your code here
    pass

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