spectral norm from SVD

How big is a matrix? Not entry-by-entry, but as a transformation — how much can it amplify a vector? The spectral norm answers this: ||A||_2 = max over ||x|| = 1 of ||Ax||. It is the largest stretch A applies to any unit vector, the worst-case gain of the map. And the SVD makes it trivial to read off: ||A||_2 = sigma_1, the largest singular value.

The reason is the geometry of the SVD. A turns the unit sphere into an ellipsoid whose semi-axes have lengths equal to the singular values; the longest semi-axis is sigma_1, achieved exactly in the direction of the first right singular vector v_1 (which lands on sigma_1 u_1). No unit input can be stretched more than that, and v_1 hits the maximum, so the operator norm is precisely sigma_1.

This is the operator norm induced by the Euclidean norm on vectors, hence the subscript 2. It is genuinely a norm on matrices — it obeys the triangle inequality and, crucially, it is submultiplicative: ||AB||_2 <= ||A||_2 ||B||_2. That submultiplicativity is what makes it the workhorse for bounding error propagation through chains of linear operations.

Contrast it with the Frobenius norm, which sums the squares of all singular values; the spectral norm cares only about the single largest. So the spectral norm measures worst-case stretch, while Frobenius measures total energy. For sensitivity and stability analysis you usually want the spectral norm, since it is the tight bound on how much A can blow up an input.