Matrix inversion cases with size-independent tensor rank estimates

Let MM be a matrix of order n=pqn=pq. Then the tensor rank of MM is defined as the minimal possible ρρ in expressions of the form M=∑t=1ρUt⊗Vt, where UtUt and VtVt are matrices of order pp and qq, respectively. Let MM be a nonsingular matrix of tensor rank 3 and, moreover, of the formM=I+A⊗X+Y⊗BM=I+A⊗X+Y⊗Bwith rankX=rankY=1. Then, it is discovered and proved that the tensor rank of M-1M-1 is bounded from above by 5 independently of pp and qq, the estimate being sharp. Some related and extended results are also given.