Abstract
The tensor rankA of the linear spaceA generated by the set of linearly independent matricesA 1, A2, …, Ap, is the least integert for wich there existt diadsu (r) v (r)τ, τ=1,2,...,t, such that\(A_s = \mathop \sum \limits_{r = 1}^t w_{s,r} u^{(r)} v^{(r)T} \). IfA=n+k,k≪n then some computational problems concerning matricesA∈A can be solyed fast. For example the parallel inversion of almost any nonsingular matrixA∈A costs 3 logn+0(log2 k) steps with max(n 2+p (n+k), k2 n+nk) processors, the evaluation of the determinant ofA can be performed by a parallel algorithm in logp+logn+0 (log2 k) parallel steps and by a sequential algorithm inn(1+k 2)+p (n+k)+0 (k 3) multiplications. Analogous results hold to accomplish one step of bisection method, Newton's iterations method and shifted inverse power method applied toA−λB in order to compute the (generalized) eigenvalues provided thatA, B∈A. The same results hold if tensor rank is replaced by border rank. Applications to the case of banded Toeplitz matrices are shown.
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Dedicated to Professor S. Faedo on his 70th birthday
Part of the results of this paper has been presented at the Oberwolfach Conference on Komplexitatstheorie, November 1983
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Bini, D. Tensor and border rank of certain classes of matrices and the fast evaluation of determinant inverse matrix and eigenvalues. Calcolo 22, 209–228 (1985). https://doi.org/10.1007/BF02576204
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DOI: https://doi.org/10.1007/BF02576204