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nhigham.com | ||
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djalil.chafai.net
Let $X$ be an $n\times n$ complex matrix. The eigenvalues $\lambda_1(X), \ldots, \lambda_n(X)$ of $X$ are the roots in $\mathbb{C}$ of its characteristic polynomial. We label them in such a way that $\displaystyle |\lambda_1(X)|\geq\cdots\geq|\lambda_n(X)|$ with growing phases. The spectral radius of $X$ is $\rho(X):=|\lambda_1(X)|$. The singular values $\displaystyle s_1(X)\geq\cdots\geq s_n(X)$ of $X$ are the eigenvalues of the positive semi-definite Hermitian... |
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www.ethanepperly.com
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djalil.chafai.net
This post is devoted to certain properties of random permutation matrices. Let \( {\mathfrak{S}_n} \) be the symmetric group, \( {n\geq2} \), and let \( {\mathcal{P}_n} \) be the group of \( {n\times n} \) permutation matrices, obtained by the isomorphism \( {\sigma\in\mathfrak{S}_n\mapsto P_\sigma={(\mathbf{1}_{j=\sigma(i)})}_{1\leq i,j\leq n}} \). We know that \( {\mathcal{P}_n} \) is a subgroup of the orthogonal group... |
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bldavies.com
Let \(A\) be the \(n\times n\) matrix with \({ij}^\text{th}\) entry \(A_{ij}=\min\{i,j\}\). From a previous post, we know \(A\) has a tridiagonal inverse \(A^{-1}\) with \({ij}^\text{th}\) entry1 $$\left[A^{-1}\right]_{ij}=\begin{cases} 2 & \text{if}\ i=j |
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