A note on matrix rigidity

In this paper we give an explicit construction of n × n matrices over finite fields which are somewhat rigid, in that if we change at most k entries in each row, its rank remains at least Cn(logq k)/k, where q is the size of the field and C is an absolute constant. Our matrices satify a somewhat stronger property, we which explain and call “strong rigidity.” We introduce and briefly discuss strong rigidity, because it is in a sense a simpler property and may be easier to use in giving explicit constructions. Recently there has been interest in giving explicit constructions of n × n matrices which are “rigid,” in the sense that their rank is high and remains high when a few of their coefficients are changed (see [Val77], [Gri76], [Raz], and [PSR]). It is easy to construct n× n matrices over infinite fields, F, such that when no more that k of the entries of each row are altered, the rank remains at least n/k; one can take a van der Monde matrix, for example. In this note we give an explicit construction of a matrix which is slightly more rigid than such a construction, for finite fields, F, and k larger than some constant (depending on the size of the field). These matrices are actually “strongly rigid,” in a sense that we will discuss later. ∗This paper was written while on leave from Princeton, at the Hebrew University. The author wishes to acknowledge the National Science Foundation for supporting this research in part under PYI grant CCR–8858788, and a grant from the program of Medium and Long Term Research at Foreign Centers of Excellence. Pudlak and Savitzky have shown that over the real numbers, a Hadamard matrix of dimension n remains of rank r if no more than n/ ( r log r ) of its entries are changed. Razborov has improved this to n/ ( r log r ) .