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Let $A$ be a random $m$ by $n$ rectangular sign matrix, chosen uniformly at random, with $m < n$. Let $B = A^T A$. We know, for example, that $B$ is a square and symmetric $n$ by $n$ matrix with all …

If $A$ is chosen uniformly at random over all possible $n$ by $n$ Toeplitz (or circulant) (0,1)-matrices, can we give any bounds for the expected size of the determinant of $AA^T$? All arithmetic is …

Consider Toeplitz matrices where the entries in the first row and column (which define the whole matrix) are independently chosen to be either $1$ or $0$ with probability $1/2$. Define $p_n$ to be the …

Consider a random $m$ by $n$ matrix $M$ with $m \leq n$, chosen uniformly over all those whose elements are in $\{0,1\}$ (or $\{-1,1\}$ if it is any easier). Is there any mathematical theory that …

Consider a random $m$ by $n$ partial circulant matrix $M$ whose entries are chosen independently and uniformly from $\{0,1\}$ and let $m < n$. Now consider a random $n$ dimensional vector $v$ whose e …

If $A$ is chosen uniformly at random over all possible $m \times n$ (0,1)-matrices, what is the expected size of the absolute value of the determinant of $AA^T$. We can assume $m < n$ and all arithmet …