By Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki

This ebook presents complete summaries of theoretical (algebraic) and computational facets of tensor ranks, maximal ranks, and usual ranks, over the genuine quantity box. even if tensor ranks were frequently argued within the advanced quantity box, it may be emphasised that this ebook treats genuine tensor ranks, that have direct functions in facts. The booklet offers numerous attention-grabbing rules, together with determinant polynomials, determinantal beliefs, completely nonsingular tensors, completely complete column rank tensors, and their connection to bilinear maps and Hurwitz-Radon numbers. as well as stories of easy methods to make certain actual tensor ranks in information, international theories similar to the Jacobian process also are reviewed in info. The ebook comprises to boot an available and complete creation of mathematical backgrounds, with fundamentals of confident polynomials and calculations by utilizing the Groebner foundation. in addition, this e-book presents insights into numerical tools of discovering tensor ranks via simultaneous singular worth decompositions.

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**Example text**

Ap−1 ) + rank(Ap ) n(p − 2) . ≤ 2m + 2 For a subset of U of TR (f ), cl U and int U denote the Euclidean closure and interior of U , respectively. The border rank (denoted by brank K (T )) of a tensor T over K is the minimal integer r such that there exist tensors Tn , n ≥ 1 with rank r that converge to T as n goes to infinity. By definition, we see that brank K (T ) ≤ rank K (T ). 1) Let K be an algebraically closed field or R, and (A; B; C) ∈ TK (n, n, 3). If A is nonsingular, then 1 brank K (A; B; C) ≥ n + rank(BA−1 C − CA−1 B).

Xt ) denotes the tensor ⎛⎛ A1 ⎜⎜ A2 ⎜⎜ ⎜⎜ .. ⎜⎜ . ⎝⎝ O ⎞ ⎛ O ⎟⎟⎟ ; ⎜⎜⎜ ⎟ ⎜ ⎠ ⎝ At ⎞⎞ B1 B2 O ⎟⎟ ⎟⎟ O ⎟⎟ . ⎟⎟ ⎠⎠ 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.

Au , Cu+1 , . . , Cr . We write r Ak = αki Ci (1 ≤ k ≤ fn ) and i=1 u Cj = r−u βjh Ah + h=1 γjh Ch+u (1 ≤ j ≤ u) h=1 for some (αki )k,i ∈ TK (fn , r), (βjh )j,h ∈ TK (u, u), (γjh )j,h ∈ TK (u, r − u). We see that u Ak − h=1 ⎛ ⎝ u ⎞ αkj βjh ⎠ Ah = j=1 ⎛ r ⎝αkh + h=u+1 u ⎞ αkj γj,h−u ⎠ Ch j=1 for u + 1 ≤ k ≤ fn . Put P = (αki )u