Abstract
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,...,Ae, denoted rk(A1,...,Ae), 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 ℂn 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.
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Literature cited
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Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 47, pp. 159–163, 1974.
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Grigor'ev, D.Y. Algebraic complexity of the computation of a pair of bilinear forms. J Math Sci 9, 264–268 (1978). https://doi.org/10.1007/BF01578549
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DOI: https://doi.org/10.1007/BF01578549