Abstract
Let M n (F) denote the set of n-by-n matrices over the field F. We consider the following question: Among matrices A ∈ M n (F) with rank A = k < n, how many diagonal entries of A must be changed, at worst, in order to guarantee that the rank of A is increased. Our initial motivation arose from an error pointed out in [BOvdD], but we also view this problem as intrinsically important. The simplest example that shows that one entry does not suffice is the familiar Jordan block \( \left[ \begin{gathered} 0 1 \hfill \\ 0 0 \hfill \\ \end{gathered} \right]. \)
This manuscript was prepared while the first three authors were visitors at the Institute for Mathematics and its Applications, Minneapolis, Minnesota. The research of C.R. Johnson and R. Loewy was supported by grant No. 90–00471 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
H. Bart, I. Gohberg and M.A. Kaashoek, The coupling method for solving integral equations, in: Topics in Operator Theory, Systems and Networks, The Rehovot Workshop (Eds: H. Dym, I. Gohberg), Operator Theory: Adv. Appl. OT 12, Birkhäuser Verlag, Basel (1984), pp. 39–73.
W.W. Barrett, D.D. Olesky and P. van den Driessche, A Note on Eigenvalues of Fixed Rank Perturbations of Diagonal Matrices, Linear and Multilinear Algebra 30 (1991), pp. 13–16.
F.G. Frobenius, Über zerlegbare Determinanten, in: F.G. Frobenius, Gesammelte Abhandlungen, Band III, Springer-Verlag Berlin-Heidelberg (1968), pp. 701–702.
M. Fiedler, T.L. Markham, Completing a Matrix when Certain Entries of its Inverse are Specified, Linear Algebra and its Applications, 74 (1986), pp. 225–237.
W.H. Gustafson, A Note on Matrix Inversion, Linear Algebra and its Applications 57 (1984), pp. 71–73.
R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, New York, NY, 1985.
C.R. Johnson and M. Lundquist, Operator Matrices with Chordal Inverse Patterns, IMA Preprint Series # 885, October 1991.
L. Mirsky and H. Perfect, Systems of Representatives, Journal of Mathematical Analysis and Applications 15 (1966), pp. 520–568.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1993 Springer-Verlag New York, Inc.
About this paper
Cite this paper
Barrett, W., Johnson, C.R., Loewy, R., Shalom, T. (1993). Rank Incrementation via Diagonal Perturbations. In: Brualdi, R.A., Friedland, S., Klee, V. (eds) Combinatorial and Graph-Theoretical Problems in Linear Algebra. The IMA Volumes in Mathematics and its Applications, vol 50. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8354-3_9
Download citation
DOI: https://doi.org/10.1007/978-1-4613-8354-3_9
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4613-8356-7
Online ISBN: 978-1-4613-8354-3
eBook Packages: Springer Book Archive