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Histogram of eigenvalues for one hundred 40 × 40 random circulant Hankel matrices. A symmetrized Rayleigh distribution is shown in red.
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We introduce a new matrix operation on a pair of matrices, $\text{swirl}(A,X),$ and discuss its implications on the limiting spectral distribution. In a special case, the resultant ensemble converges almost surely to the Rayleigh distribution. In proving this, we provide a novel combinatorial proof that the random matrix ensemble of circulant Hanke...
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Context 1
... 2: We will show that the row vectors corresponding to odd-odd pairs of indices cannot be part of any nonempty linear combination of row vectors summing to zero (the proof follows for even-even pairs as well). Suppose indices i r and i r+2i are paired for r odd and i ≥ 1. The corresponding row vector is of the form (0, 0, . . . , 1, 1, 0, . . . , 0, −1 − 1, 0, . . . , 0) with 1's in the r th and (r + 1) st indices and −1's in the (r + 2i) th and (r + 2i + 1) st indices. As before, to cancel out the contribution of i r , we need i r to appear as a second index and contribute negatively. As a result, i r−1 must appear as a first index and contribute negatively. Then we need i r−1 to appear as a ...
Context 2
... 2: We will show that the row vectors corresponding to odd-odd pairs of indices cannot be part of any nonempty linear combination of row vectors summing to zero (the proof follows for even-even pairs as well). Suppose indices i r and i r+2i are paired for r odd and i ≥ 1. The corresponding row vector is of the form (0, 0, . . . , 1, 1, 0, . . . , 0, −1 − 1, 0, . . . , 0) with 1's in the r th and (r + 1) st indices and −1's in the (r + 2i) th and (r + 2i + 1) st indices. As before, to cancel out the contribution of i r , we need i r to appear as a second index and contribute negatively. As a result, i r−1 must appear as a first index and contribute negatively. Then we need i r−1 to appear as a ...
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We introduce a new matrix operation on a pair of matrices, sw(A,X), and discuss its implications on the limiting spectral distribution. In a special case, the resultant ensemble converges almost surely to the Rayleigh distribution. In proving this, we provide a novel combinatorial proof that the random matrix ensemble of circulant Hankel matrices c...
Let $f,g_1,\dots,g_m$ be polynomials with real coefficients in a vector of variables $x=(x_1,\dots,x_n)$. Denote by $\text{diag}(g)$ the diagonal matrix with coefficients $g=(g_1,\dots,g_m)$ and denote by $\nabla g$ the Jacobian of $g$. Let $C$ be the set of critical points defined by \begin{equation} C=\{x\in\mathbb R^n\,:\,\text{rank}(\varphi(x))...
