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Example text

Property (ii) gives some clue to the strange name ‘geometric orthogon⊥ ality’. The space W ∩ (W ∩ U ) is the orthogonal complement of W ∩ U ⊥ inside W , and U ∩ (W ∩ U ) is the orthogonal complement of W ∩ U inside U . So geometric orthogonality means that these two complements are orthogonal to each other as well as to W ∩ U : ⊥ W ∩ (W ∩ U ) ⊥ ⊥ U ∩ (W ∩ U ) . Thus subspaces W and U are geometrically orthogonal if they are as orthogonal as they can be given that they have a non-zero intersection.

If M1 ∈ F Γ×∆ and M2 ∈ F ∆×Φ then M1 M2 is the matrix in F Γ×Φ defined by (M1 M2 )(γ, φ) = M1 (γ, δ)M2 (δ, φ). δ∈∆ All the usual results about matrix multiplication hold. In particular, matrix multiplication is associative, and (M1 M2 ) = M2 M1 . Similarly, if M ∈ F Γ×∆ then M defines a linear transformation from ∆ F to F Γ by f → Mf where (M f )(γ) = M (γ, δ)f (δ) for γ ∈ Γ. δ∈∆ If Φ is any subset of Γ × ∆ then its characteristic function χΦ satisfies χΦ (γ, δ) = 1 if (γ, δ) ∈ Φ 0 otherwise. In the special case that Γ = ∆ = Ω we call χΦ the adjacency matrix of Φ and write it AΦ .

The reader who is familiar with group theory will realise that, throughout this subsection, the integers modulo n may be replaced by any finite Abelian group, so long as some of the detailed statements are suitably modified. ) We can define multiplication on F Ω by χα χβ = χα+β extended to the whole of F Ω by  λα χα α∈Ω for α and β in Ω,  µβ χβ  =  λα µβ χα+β . α∈Ω β∈Ω β∈Ω Thus, if f and g are in F Ω then (f g)(ω) = f (α)g(ω − α) α∈Ω so this multiplication is sometimes called convolution. It is associative (because addition in Zn is), is distributive over vector addition, and commutes with scalar multiplication, so it turns F Ω into an algebra, called the group algebra of Zn , written F Zn .

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