The rigidity of a matrix $A$ for target rank $r$ is the minimum number

of entries of $A$ that must be changed to ensure that the rank of

the altered matrix is at most $r$. Since its introduction by Valiant

\cite{Val77}, rigidity and similar rank-robustness functions of

matrices have found numerous applications in circuit complexity,

communication complexity, and learning complexity. Almost all $\nbyn$

matrices over an infinite field have a rigidity of $(n-r)^2$. It is a

long-standing open question to construct infinite families of

\emph{explicit} matrices even with superlinear rigidity when $r=\Omega(n)$.

In this paper, we construct an infinite family of complex matrices

with the largest possible, i.e., $(n-r)^2$, rigidity. The entries of

an $\nbyn$ matrix in this family are distinct primitive roots of unity

of orders roughly \SL{$\exp(n^4 \log n)$}. To the best of our knowledge, this is

the first family of concrete (but not entirely explicit) matrices

having maximal rigidity and a succinct algebraic description.

Our construction is based on elimination theory of polynomial

ideals. In particular, we use results on the existence of polynomials

in elimination ideals with effective degree upper bounds (effective

Nullstellensatz). Using elementary algebraic geometry, we prove that

the dimension of the affine variety of matrices of rigidity at

most $k$ is exactly $n^2 - (n-r)^2 +k$. Finally, we use elimination theory to

examine whether the rigidity function is semicontinuous.