By Daniel Alpay

This publication presents the principles for a rigorous idea of sensible research with bicomplex scalars. It starts with an in depth learn of bicomplex and hyperbolic numbers after which defines the thought of bicomplex modules. After introducing a couple of norms and internal items on such modules (some of which seem during this quantity for the 1st time), the authors strengthen the speculation of linear functionals and linear operators on bicomplex modules. All of this can serve for lots of varied advancements, like the traditional sensible research with complicated scalars and during this ebook it serves because the foundational fabric for the development and examine of a bicomplex model of the well-known Schur analysis.

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**Sample text**

11). 8. Let A ≤ BCn×n . The following are equivalent: (1) A is hyperbolic positive. (2) A = B ∗ t · B where B ≤ BCm×n for some m ≤ N. (3) A = C 2 where the matrix C is hyperbolic positive. 9), and assume that (1) holds. 12) where U and V are matrices in Cn×n (i). Thus A = B ∗ t B with B := U e + V e† , so that (2) holds with m = n. Assume now that (2) holds with B ≤ BCm×n for some m ≤ N. Writing B = U e + V e† , where now U and V belong to Cm×n (i) we have t t A = U U e + V V e† , and so (1) holds.

12), it follows that a sequence {Z n }n∈N converges to the bicomplex number Z 0 with respect to the hyperbolic-valued norm if and only if it converges to Z 0 with respect to the Euclidean norm, and so even though the two norms cannot be compared as they take values in different rings, one still obtains the same notion of convergence. It is possible to give a precise geometrical description of the set of bicomplex numbers with a fixed hyperbolic norm, that is, we want to introduce the “sphere of hyperbolic radius γ0 inside BC”.

This concludes the proof. ⊗ Analogously, one can use the idempotent representation of a bicomplex matrix to determine its invertibility. 3. Let A = A1,i e + A2,i e† = A1, j e + A2, j e† ≤ BCn×n , A1,i , A2,i ≤ Cn×n (i), A1, j , A2, j ≤ Cn×n (j) be a bicomplex matrix. Then A is invertible if and only if A1,i and A2,i are invertible in Cn×n (i) and A1, j , A2, j are invertible in Cn×n ( j). Proof: A matrix A is invertible if and only if there exists B = B1,i e + B2,i e† ≤ BCn×n such that AB = B A = I .