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numerical-range:generalizations:joint-numerical-range

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 numerical-range:generalizations:joint-numerical-range [2019/10/01 13:02]rkukulski numerical-range:generalizations:joint-numerical-range [2020/06/30 07:28] (current)plewandowska Both sides previous revision Previous revision 2020/06/30 07:28 plewandowska 2020/05/16 13:03 plewandowska 2020/05/16 13:00 plewandowska 2020/05/16 12:58 plewandowska 2020/05/16 12:58 plewandowska 2020/05/16 12:57 plewandowska 2020/05/16 12:56 plewandowska 2020/05/16 12:56 plewandowska 2020/05/16 12:53 plewandowska 2020/05/16 12:52 plewandowska 2020/05/16 12:50 plewandowska 2020/05/16 12:50 plewandowska 2020/05/16 12:49 plewandowska 2020/05/16 12:48 plewandowska 2020/05/16 12:47 plewandowska 2020/05/16 12:47 plewandowska 2020/02/19 10:40 plewandowska 2019/10/01 13:02 rkukulski 2019/10/01 13:01 plewandowska 2019/10/01 12:55 plewandowska 2018/11/04 09:56 lpawela 2018/11/04 09:48 lpawela 2018/10/08 08:42 plewandowska [Definition] 2018/10/08 08:22 plewandowska [Definition] 2018/06/15 16:20 lpawela 2018/06/15 16:17 lpawela 2018/06/06 12:09 lpawela 2018/06/06 12:02 lpawela 2018/05/24 13:33 lpawela 2018/05/24 13:30 lpawela 2018/05/24 10:24 lpawela 2018/05/24 10:23 lpawela 2018/05/24 10:21 lpawela created Next revision Previous revision 2020/06/30 07:28 plewandowska 2020/05/16 13:03 plewandowska 2020/05/16 13:00 plewandowska 2020/05/16 12:58 plewandowska 2020/05/16 12:58 plewandowska 2020/05/16 12:57 plewandowska 2020/05/16 12:56 plewandowska 2020/05/16 12:56 plewandowska 2020/05/16 12:53 plewandowska 2020/05/16 12:52 plewandowska 2020/05/16 12:50 plewandowska 2020/05/16 12:50 plewandowska 2020/05/16 12:49 plewandowska 2020/05/16 12:48 plewandowska 2020/05/16 12:47 plewandowska 2020/05/16 12:47 plewandowska 2020/02/19 10:40 plewandowska 2019/10/01 13:02 rkukulski 2019/10/01 13:01 plewandowska 2019/10/01 12:55 plewandowska 2018/11/04 09:56 lpawela 2018/11/04 09:48 lpawela 2018/10/08 08:42 plewandowska [Definition] 2018/10/08 08:22 plewandowska [Definition] 2018/06/15 16:20 lpawela 2018/06/15 16:17 lpawela 2018/06/06 12:09 lpawela 2018/06/06 12:02 lpawela 2018/05/24 13:33 lpawela 2018/05/24 13:30 lpawela 2018/05/24 10:24 lpawela 2018/05/24 10:23 lpawela 2018/05/24 10:21 lpawela created Line 7: Line 7: L(A_1,​\ldots,​A_k) = \left\{ \left( \Tr \rho A_1,\ldots, \Tr \rho A_k \right): \rho \in \Omega_d \right\}. L(A_1,​\ldots,​A_k) = \left\{ \left( \Tr \rho A_1,\ldots, \Tr \rho A_k \right): \rho \in \Omega_d \right\}.  - Due to taking all mixed states the joint numerical range is a convex body in a $k$-dimensional space. + Due to taking all mixed states the joint numerical range is a convex body in a $k$-dimensional space. ​More information about convexity of joint numerical range we can find in [( :​li2020joint )]. + + ===== Kippenhahn Theorem'​s for joint numerical range ===== + An algebraic curve has been associated with the numerical range and has been + studied from the 1930s on [( :​plaumann2019kippenhahn )]. Let $\1$ denote the $d\times d$ identity matrix, + $+ p(x)=\det( x_0 \1 + x_1 A_1 + \cdots + x_n A_n ), +$ + and consider the complex projective hypersurface + $+ \mathcal{V}(p)=\{x\in\mathbb{P}^n \mid p(x) = 0 \}. +$ It was shown  that the eigenvalues of + $A_1+\ii A_2$ are the foci of the affine + curve of real points + $+ T=\{ y_1+\ii y_2 \mid y_1,​y_2\in\mathbb{R},​(1:​y_1:​y_2)\in \mathcal{V}(p)^\ast\}\subset\mathbb{C}. +$ + Kippenhahn ​ recognized the meaning of the convex hull of $T$. + + + ==== Theorem ==== + The numerical range of $A_1+\ii A_2$ is the convex hull of $T$, in other words, + $W(A_1+\ii A_2)=\text{conv}(T)$. + + + + Let we denote $T_i$ as the semi-algebraic set + $+ T_i = \left\{ (y_1,​\dots,​y_n) \in (\mathbb{R}^n)^\ast \mid + (1:​y_1:​\dots:​y_n)\in (X_i^\ast)_\text{reg} \right\}, + \qquad i=1,\dots,r +$ and $(X^{*}_i)_{\text{reg}}$ ​ as the set of the regular points of the dual variety $X^*$. + + ==== Theorem ==== + The joint numerical range $W$ is the convex hull of the Euclidean closure of + $T_1\cup\cdots\cup T_r$, in other words, $W = \text{conv}\text{cl}(T_1\cup\cdots\cup T_r)$. + + ==== Classification of JNRs ==== ==== Classification of JNRs ==== Line 32: Line 70: - If two segments are present in the boundary, there exist an infinite number o other segments. - If two segments are present in the boundary, there exist an infinite number o other segments. - All configurations permitted by these rules are realized. Let us denote by $e$ the number of ellipses in the boundary and by $s$ the number of segments. There exist object with: + All configurations permitted by these rules are realized. Let us denote by $e$ the number of ellipses in the boundary and by $s$ the number of segments. There exists ​object with [( :​xie2019observing )]: * no flat parts in boundary at all $e=0$, $s=0$, {{ :​numerical-range:​generalizations:​img_5903.png?​nolink&​200 |}} * no flat parts in boundary at all $e=0$, $s=0$, {{ :​numerical-range:​generalizations:​img_5903.png?​nolink&​200 |}} * one segment in the boundary $e=0$, $s=1$, {{ :​numerical-range:​generalizations:​img_5935.png?​nolink&​200 |}} * one segment in the boundary $e=0$, $s=1$, {{ :​numerical-range:​generalizations:​img_5935.png?​nolink&​200 |}} Line 47: Line 85: ====Application==== ====Application==== - An example application of numerical range can be found in [( :​szymanski2019geometric )], [( :​czartowskiseparability )]. + An example application of numerical range can be found in [( :​szymanski2019geometric )], [( :​czartowskiseparability ​)], [( :​plaumann2019kippenhahn ​)].
numerical-range/generalizations/joint-numerical-range.1569934964.txt.gz · Last modified: 2019/10/01 13:02 by rkukulski