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It is proved that two diagonal matrices diag(a₁,…,a_n) and diag(b₁,…,b_n) over a local ring R are equivalent if and only if there are two permutations σ,τ of {1,2,…,n} such that [Formula: see text] and [Formula: see text] for every i=1,2,…,n. Here [R/aR]_e denotes the epigeny class of R/aR, and [R/aR]_l denotes the lower part of R/aR. In some particular cases, like for instance in the case of R local commutative, diag(a₁,…,a_n) is equivalent to diag(b₁,…,b_n) if and only if there is a permutation σ of {1,2,…,n} with a_iR=b_σ(i)R for every i=1,…,n. These results are obtained studying the direct-sum decompositions of finite direct sums of cyclically presented modules over local rings. The theory of these decompositions turns out to be incredibly similar to the theory of direct-sum decompositions of finite direct sums of uniserial modules over arbitrary rings.