 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ib/paper-4/11g/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

m \times n

A

R

D = U A V = diag (d_{1}, \dots, d_{r}, 0, \dots, 0), d_{1} ∣ d_{2} ∣ \dots ∣ d_{r},

where

U \in G L_{m} (R)

and

V \in G L_{n} (R)

. It is guaranteed to exist for every matrix over a principal ideal domain. The entries

d_{i}

are determined up to multiplication by units; over

Z

they are conventionally chosen positive.

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