# Numerical Shadow

The web resource on numerical range and numerical shadow

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numerical-range:animations:ginibre

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 numerical-range:animations:ginibre [2013/03/16 12:06]gawron created numerical-range:animations:ginibre [2018/05/22 14:17] (current)plewandowska 2018/05/22 14:17 plewandowska 2018/05/22 14:16 plewandowska 2013/04/16 00:29 lpawela animations:ginibre renamed to numerical-range:animations:ginibre2013/03/16 13:24 gawron 2013/03/16 13:22 gawron 2013/03/16 13:22 gawron 2013/03/16 12:06 gawron created Next revision Previous revision 2018/05/22 14:17 plewandowska 2018/05/22 14:16 plewandowska 2013/04/16 00:29 lpawela animations:ginibre renamed to numerical-range:animations:ginibre2013/03/16 13:24 gawron 2013/03/16 13:22 gawron 2013/03/16 13:22 gawron 2013/03/16 12:06 gawron created Line 1: Line 1: + ====== Numerical range and spectrum of random Ginibre matrix ====== + + Let $G$ be a matrix of $\mathrm{dim}\ G=1000$ drawn from Ginibre ensemble and let $G_d$ be a family of matrices such that + $G_d=P_d(T)$,​ where $T$ is upper triangular matrix obtained by Schur decomposition of $G$ such that $G=UTU^\dagger$. $P_d$ are orthogonal projections $P_d(\cdot)=\sum_{i=1}^d \ket{i}\bra{l_i}\cdot\ket{l_i}\bra{i}$,​ where $l_i$ is a sequence of integers from $1$ to $1000$. $G_d$ are normalized so $\tr G_d G_d^\dagger=\mathrm{dim}\ G_d$. + + In the figure red dots indicate spectrum of $G_d$, gray area is numerical range $W(G_d)$, green circle has radius $1$ outer circle has radius $\sqrt{2}$. + {{ :​animations:​animation-ginibre.gif?​nolink |}} {{ :​animations:​animation-ginibre.gif?​nolink |}} 