Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 4 / 11G / a

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- Solution a

Solution

A Smith normal form of an

m \times n

A

over a principal ideal domain

R

is a diagonal matrix

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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