Matrix inversion cases with size-independent tensor rank estimates
Over an arbitrary field let \(M\) be a nonsingular \(pq\times pq\) matrix of the form \[ M=I+A \otimes X+Y \otimes B, \] where \(A,Y\) are \(p\times p\) and \(X,B\) are \(q\times q\). Assume additionally that rank \(X = \text{rank } Y = 1\) and that \(I+A\otimes X\) and \(I+Y\otimes B\) are non-singular. It is shown that \(M^{-1}\) can be written in the form \[ M^{-1}=I +\hat A\otimes X +Y\otimes \hat B + Y_1\otimes X_1 + Y_2\otimes X_2 \] for some matrices \(\hat A, \hat B\) and matrices \(X_1,X_2,Y_1,Y_2\) of rank at most \(1\). This implies that the tensor rank of \(M^{-1}\) is bounded from above by 5 independently of \(p\) and \(q\). For matrices over the complex numbers a stronger version of the result is established.
scientific article; zbMATH DE number 5574547