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Tao and Vu (1) have shown that the distribution of the smallest singular value of a random matrix is "universal", i.e. independent of the particular random variable populating the matrix. They are in …

There are only $2^{16}$ possible $4 \times 4$ matrices over $\mathbb{F}_2$. That means that by examining $2^{32}$ possibilities, you can exhaust $A$ and $B$. Once $A$ and $B$ are known, your first e …

Take a function $f: Z_N\rightarrow R$. Construct an $N \times N$ matrix where the $(i,j)$th element of the matrix is $f(i-j)$, where $i-j$ is interpreted mod $Z_N$. The resulting matrices are precis …

This is a little numerical experiment, not an answer, but it provides a hypothesis for the shape of a solution. We (computationally) confirmed that matrices with this form have a discrepancy of $(n-2) …

Consider a field $F$. Suppose we have two vectors $a,b\in F^n$, and an invertible matrix $G\in F^{n\times n}$. Let $c\in F^n$ be the point-wise product of $a$ and $b$, that is, $c_i=a_ib_i$. Let $x …

Consider $S$, the set of all $n\times m$ real matrices with specified row sums $(r_1,...,r_n)$, column sums $(c_1,...,c_m)$, and strictly positive entries. For any matrix $A$, define $$ D_A(i,j)=\fra …

A non-zero complex number can be uniquely written in polar form as $re^{i\theta}$. There is an analogous result for complex matrices: any invertible complex matrix can be uniquely written as $UP$, wh …

If you will forgive the self-serving pointer, I wrote a little paper examining this question from the perspective of "what's the sparsest basis of eigenvectors inside the DFT?". The paper's introduct …