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By Alex Figotin; Ilya Vitebskiy

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0  S0−1     . .. ... 1   .. 0 ··· 0 0 1 Slow light in photonic crystals 55 for an invertible n × n matrix S0 . In other words, there exists a basis f0 , f1 , . . , fn such that −1 −1 ζ −1 0 T0 f0 = f0 , ζ 0 T0 f1 = f1 + f0 , · · · , ζ 0 T0 fn = fn + fn−1 . (269) The basis f0 , f1 , . . , fn reducing T0 to its canonical form is not unique. What is unique is the following set of spans Span {f0 } , Span {f0 , f1 } , Span {f0 , f1 , . . , fn−1 } . (270) Possible bases preserving the canonical matrix to the right of (269) are described by the following transformations    1 1 0 ··· 1 1 0 ··· 0    .

D ˆ (r) H (76) All variables in Eqs. (75) and (76) are frequency dependent. In lossless media, the material tensors ˆε (r) and µ ˆ (r) are Hermitian In lossless media: ˆε (r) = ˆε† (r) , µ ˆ (r) = µ ˆ † (r) , (77) where the dagger † denotes the Hermitian conjugate. In lossless non-magnetic media, both tensors ˆε and µ ˆ are also real and symmetric In lossless non-magnetic media: ˆε (r) = ˆε∗ (r) = ˆεT (r) , µ ˆ (r) = µ ˆ ∗ (r) = µ ˆ T (r) , (78) Slow light in photonic crystals 31 where the asterisk denotes the complex conjugate and the superscript T denotes matrix transposition.

161) 0 j2 (162) Let us introduce j2 = 0 1 −1 0 , J= −j2 0 = j2 ⊗ j2 , and notice that JA(0) = 0 j2 a(0) −j2 a(0) 0 = 0 −1 1 0 ⊗ j2 a(0) = j2 ⊗ j2 a(0) . (163) Slow light in photonic crystals 43 Recall the basic properties of the tensor product operation: if A and B are square matrices and u and v are vectors of related dimensions then [A ⊗ B] (u ⊗ v) = (Au) ⊗ (Bv) , (u1 ⊗ v1 , u2 ⊗ v2 ) = (u1 , u2 ) (v1 , v2 ) . (164) Suppose now that we know the set of eigenvectors and eigenvalues for two square matrices A and B, namely Auj = λj uj , Bvm = µm vm .

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