By Franklin Mendivil

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**Extra resources for Fractals, Graphs, and Fields**

**Sample text**

A permutation matrix P = [pij ] of order m is a square matrix that has exactly one 1 in each row and column and 0’s elsewhere. Thus a permutation matrix of order m has exactly m nonzero entries and each of these m entries equals 1. Permutation matrices correspond to permutations in the following way: Let σ = k1 k2 . . km be a permutation of {1, 2, . . , m}. Let P = [pij ] be the square matrix of order m defined by 1, 0, pij = if j = ki , otherwise. Then P is a permutation matrix and every permutation matrix of order m corresponds to a permutation of {1, 2, .

Nνλ . Assume also that the column partition of A agrees with the row partition of B. This means that Mik is an mi by nk matrix and Nkj is an nk by pj matrix. Here the integers m, n, and p are partitioned as m = m1 + m2 + · · · + mµ , n = n1 + n2 + · · · + nν , and p = p1 + p2 + · · · + pλ . Under these circumstances, we say that A and B are conformally partitioned. Let the set of black vertices of G(A) be partitioned in accordance with the partition of the integer m, and let the set of white vertices of G(A) be partitioned according to the partition of the integer n.

Congruence satisfies a basic property with regard to addition and mutltiplication that is easily verified: If a ≡ b (mod m) and c ≡ d (mod m), then a + c ≡ b + d (mod m) and ac ≡ bd (mod m). This property allows one to add and multiply equivalence classes unambiguously as follows: [a]m + [b]m = [a + b]m and [a]m · [b]m = [ab]m . Let Zm = {0, 1, 2, . . , m − 1}. Then Zm contains exactly one element from each equivalence class, and we can regard addition and multiplication of equivalence classes as addition and multiplication of integers in Zm .