Algebraic complexity of the computation of a pair of bilinear forms

The problem mentioned in the titled reduces to the estimation of the rank of a collection of matrices. The rank of a collection of matrices A₁,...,A_e, denoted rk(A₁,...,A_e), is the least number of such one-dimensional matrices that their linear combinations will represent each matrix of the given collection. For an operator A on ⁿ there exists a space V and a diagonal operator B such that; we denote the minimal dimension of such spaces V by d(V)Theorem. For any matrix A we have the equality rk (E,A.)=n+d(A), where E is the identity matrix.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 47, pp. 159–163, 1974.