Abstract
In this chapter we extend the notion of rank numbers introduced in Chapter 4 to wider sets of submatrices. Lower rank numbers for a square matrix relative to the diagonal i – j are introduced as the ranks of the maximal submatrices entirely located under that diagonal, and the upper rank numbers relative to a diagonal are defined correspondingly. If the given matrix is invertible, a strong link exists between these numbers for the matrix and its inverse. In particular, the lower and upper rank numbers relative to the main diagonal are the same for a matrix with square blocks on the main diagonal and for its inverse matrix. This implies that for such a square matrix the lower and upper quasiseparable orders coincide with the ones of the inverse matrix.
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© 2014 Springer Basel
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Eidelman, Y., Gohberg, I., Haimovici, I. (2014). Rank Numbers of Pairs of Mutually Inverse Matrices, Asplund Theorems. In: Separable Type Representations of Matrices and Fast Algorithms. Operator Theory: Advances and Applications, vol 234. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0606-0_6
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DOI: https://doi.org/10.1007/978-3-0348-0606-0_6
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Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0605-3
Online ISBN: 978-3-0348-0606-0
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