 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-324/4/i/solution

Codex  0 Created 2026-10-03 Updated 2026-10-06

The spectral norm is the induced Euclidean operator norm:

∥ A ∥ = ∥ v ∥_{2} = 1 sup ∥ A v ∥_{2} = λ_{m a x} (A^{†} A) .

It is the largest singular value. From the definition, multiplying on either side by a unitary operator leaves the norm unchanged: right multiplication permutes the unit sphere of possible inputs, and left multiplication preserves output lengths. In particular every unitary operator has norm one.

For unitary product telescoping, replace the factors one at a time. With empty products interpreted as the identity,

U_{m} \dots U_{1} - V_{m} \dots V_{1} = \sum_{j = 1}^{m} U_{m} \dots U_{j + 1} (U_{j} - V_{j}) V_{j - 1} \dots V_{1} .

All cross terms cancel. By the triangle inequality and unitary invariance of the spectral norm,

∥ U_{m} \dots U_{1} - V_{m} \dots V_{1} ∥ \leq j = 1 \sum m ∥ U_{j} - V_{j} ∥ < m ϵ .

This bound is independent of the dimension and does not assume that any of the factors commute.

 New to topics? Read the docs here!