Almost orthogonal submatrices of an orthogonal matrix

Mark Rudelson

Let $A$ be an $n \times M$ matrix whose rows are orthonormal. Let $A_I$ be a submatrix of $A$ whose column indexes belong to the set $I$. Given $\varepsilon >0$ we estimate the smallest cardinality of the set $I$, such that the operator $A_I$ is an $\varepsilon$-isometry.