Markov chains and tensor multiplications
Let \(F^ n\) denote the space of n-tuples over a field F. By an m- dimensional matrix of order n over F the author means an array \(A=[a_{\gamma}]\); \(a_{\gamma}\in F\), where \(\gamma \in \Gamma (m,n)=\{\gamma =(\gamma (1),...,\gamma (m))|\) \(1\leq \gamma (i)\leq n\), all \(i\}\). Let \(M^ m_ n(F)\) be the set of such objects. This is a vector space of dimension \(n^ m\) over F with respect to the obvious addition and scalar multiplication. One can consider \(M^ m_ n(F)\) as, say, the m-th tensor product of \(F^ n.\) In this paper two multiplications on \(M^ m_ n(F)\) are examined. The first one is non-associative and is motivated by a generalization of a Markov chain. Namely, this is defined by the formula: \[ (A.B)_{\gamma}=\sum^{n}_{x=1}a_{(\gamma (1),...,\gamma (m- 1),s)}b_{(\gamma (2),...,\gamma (m-1),s,\gamma (m))}. \] This multiplication satisfies the properties: \(A.(B+C)=A.B+A.C,\) \((A+B).C=A.C+B.C,\) it has the right multiplicative identity, it is non- associative when \(m>2\), \(n>1\), even on powers (i.e. A.(A.A)\(\neq (A.A).A\) in general). The second multiplication is defined by the formula: \((AB)=\sum^{n}_{s=1}a_{(\gamma:s)}b_{(\sigma:\gamma)},\) where \((\gamma:s)=(\gamma (1),...,\gamma (m-1),s)\) and \((s:\gamma)=(s,\gamma (2),...,\gamma (m))\) for any \(s=1,...,n\). This multiplication satisfies the properties: \(A(B+C)=AB+AC,\) \((A+B)C=AC+BC,\) it has two-sided identity, it coincides with the usual definition when \(m=2\), it satisfies \(A(BC)=A(BC).\) Spectral and unitary properties of the latter multiplication are discussed, as well as connections with block-diagonal matrix multiplication. A connection between graphs and the non-associative multiplication is exhibited.
scientific article; zbMATH DE number 4065179