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www.jeremykun.com | ||
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djalil.chafai.net
The Girko circular law theorem states that if \( {X} \) is a \( {n\times n} \) random matrix with independent and identically distributed entries (i.i.d) of variance \( {1/n} \) then the empirical measure \[ \frac{1}{n}\sum_{i=1}^n\delta_{\lambda_i(X)} \] made with the eigenvalues of \( {X} \), converges, as the dimension \( {n} \) tends to infinity, to the uniform law... |
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nhigham.com
The Cayley-Hamilton Theorem says that a square matrix $LATEX A$ satisfies its characteristic equation, that is $latex p(A) = 0$ where $latex p(t) = \det(tI-A)$ is the characteristic polynomial. This statement is not simply the substitution ``$latex p(A) = \det(A - A) = 0$'', which is not valid since $latex t$ must remain a scalar... |
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alanrendall.wordpress.com
The theorem of the title is about dividing smooth functions by other smooth functions or, in other words, representing a given smooth function in terms of products of other smooth functions. A large part of the account which follows is based on that in the book 'Normal Forms and Unfoldings for Local Dynamical Systems' by... |
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nhigham.com
In many applications a matrix $latex A\in\mathbb{R}^{m\times n}$ has less than full rank, that is, $latex r = \mathrm{rank}(A) < \min(m,n)$. Sometimes, $latex r$ is known, and a full-rank factorization $LATEX A = GH$ with $latex G\in\mathbb{R}^{m \times r}$ and $latex H\in\mathbb{R}^{r \times n}$, both of rank $latex r$, is given-especially when $latex r =... |
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