By Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki
This booklet presents complete summaries of theoretical (algebraic) and computational features of tensor ranks, maximal ranks, and regular ranks, over the genuine quantity box. even if tensor ranks were usually argued within the complicated quantity box, it may be emphasised that this publication treats actual tensor ranks, that have direct functions in records. The ebook presents numerous attention-grabbing principles, together with determinant polynomials, determinantal beliefs, totally nonsingular tensors, totally complete column rank tensors, and their connection to bilinear maps and Hurwitz-Radon numbers. as well as stories of the right way to confirm genuine tensor ranks in information, international theories equivalent to the Jacobian technique also are reviewed in information. The publication comprises besides an available and entire advent of mathematical backgrounds, with fundamentals of confident polynomials and calculations through the use of the Groebner foundation. additionally, this booklet presents insights into numerical tools of discovering tensor ranks via simultaneous singular worth decompositions.
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Extra resources for Algebraic and Computational Aspects of Real Tensor Ranks
Suppose that m ≥ 3 and let P(m, d) be the set of degree d homogeneous polynomials with coefficients in R and variables x1 , . . , xm . Since Rn×n×m → P(m, n), m T = (T1 ; . . 4 is a Zariski open subset of Rn×n×m . In fact, this inverse image is not empty by Sumi et al. 2, and therefore it is dense. Moreover, we see the following fact. 6 Suppose that 3 ≤ m ≤ n and let x1 , . . , xm be indeterminates. Then there are Euclidean open subsets O1 and O2 of Rn×n×m with the following properties. (1) O1 ∪ O2 is a dense subset of Rn×n×m in the Euclidean topology.
To discuss an upper bound of the maximal rank, we may assume that A1 , . . , Amn−k are linearly independent without loss of generality. Let X be the vector space spanned by m × n matrices A1 , . . , Amn−k . rank K (m, k, mk − k) rank-1 matrices. For 1 ≤ i ≤ m, let Yi be the vector space consisting of all matrices such that the i th row is zero if i = i. Since dim(X ∩ Yi ) = dim(X) + dim(Yi ) − dim(X ∩ Yi ) ≥ (mn − k) + n − mn = n − k, (i) of X ∩ Yi . Let Z1 , . . , we can take linearly independent matrices B1(i) , .
Rank K (m, n, mn) = mn. 1 (Atkinson and Stephens 1979, Lemma 5) Let K be a subfield of F. rank K (m, k, mk − k). Proof Let (A1 ; . . rank F (m, k, mk − k) and Bj = (Aj , O) be an m × n matrix for 1 ≤ j ≤ mk − k. Consider the tensor X = (B1 ; . . ; Bmk−k ; E1,k+1 ; . . ; E1n ; . . ; Em,k+1 ; . . ; Emn ) with format (m, n, mn − k), where Eij denotes an m × n matrix with a 1 in the (i, j) position and zeros elsewhere. rank K (m, n, mn − k) ≥ rank K (X) ≥ rank K (B1 ; . . ; Bmk−k ) + rank K (E1,k+1 ; .