By Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki

ISBN-10: 4431554580

ISBN-13: 9784431554585

ISBN-10: 4431554599

ISBN-13: 9784431554592

This publication presents accomplished summaries of theoretical (algebraic) and computational facets of tensor ranks, maximal ranks, and ordinary ranks, over the genuine quantity box. even supposing tensor ranks were frequently argued within the complicated quantity box, it may be emphasised that this booklet treats actual tensor ranks, that have direct functions in information. The ebook offers a number of fascinating principles, together with determinant polynomials, determinantal beliefs, completely nonsingular tensors, totally complete column rank tensors, and their connection to bilinear maps and Hurwitz-Radon numbers. as well as reports of how to verify actual tensor ranks in info, international theories akin to the Jacobian process also are reviewed in information. The publication contains to boot an available and entire advent of mathematical backgrounds, with fundamentals of confident polynomials and calculations through the use of the Groebner foundation. additionally, this publication presents insights into numerical tools of discovering tensor ranks via simultaneous singular worth decompositions.

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**Example text**

1 (Ja’Ja’ 1979; Sumi et al. 2009) (1) rank F (O; O) = 0 and rank F (a; 1) = 1. (2) rank F (aEk + Jk ; Ek ) = rank F (Ek ; Jk ) = k + 1 for k ≥ 2. O E = k + 1 for k ≥ 1. ; k (3) rank F ((O, Ek ); (Ek , O)) = rank F O Ek (4) rank R (Ck (c, s) + Jk ⊗ E2 ; E2k ) = 2k + 1 if s = 0 and k ≥ 1. 2 (Ja’Ja’ 1979) Let A be a tensor with format (m, n, 2) and let B be a tensor of type (D) or (E). Then, rank F (Diag(A, B)) = rank F (A) + rank F (B). In general, the rank of a tensor is not the sum of ranks of its Kronecker–Weierstrass blocks.

Bt of format (m1 + m2 + · · · + mt , n1 + n2 + · · · + nt , 2). This notation depends on the direction of slices. 1 (Gantmacher 1959, (30) in Sect. 4, XII) Let K be an algebraically closed field. A 3-tensor (A; B) ∈ TK (m, n, 2) is GL(m, K) × GL(n, K)-equivalent to a tensor of block diagonal form Diag((S1 ; T1 ), . . , (Sr ; Tr )), where each (Sj ; Tj ) is one of the following: zero tensor (O; O) ∈ TK (k, l, 2), k, l ≥ 0, (k, l) = (0, 0), (aEk + Jk ; Ek ) ∈ TK (k, k, 2), k ≥ 1, (Ek ; Jk ) ∈ TK (k, k, 2), k ≥ 1, ((O, Ek ); (Ek , O)) ∈ TK (k, k + 1, 2), k ≥ 1, O E ∈ TK (k + 1, k, 2), k ≥ 1.

Therefore by generalizing the notion of an absolutely nonsingular tensor, we arrive at the following notion. 4 Let T = (T1 ; . . ; Tm ) be a u × n × m tensor over R. T is called an Absolutely full column rank tensor if m rank =n ak Tk k=1 for any (a1 , . . , am ) ∈ Rm \{0}. 1. 3 Let u, n, and m be positive integers with u ≥ n and let T = (T1 ; . . ; Tm ) be a u × n × m tensor over R. Then the following conditions are equivalent. (1) T is Absolutely full column rank. (2) For any a = (a1 , . .

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