The web resource on numerical range and numerical shadow

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 numerical-shadow:examples:2x2 [2013/03/13 12:31]lpawela numerical-shadow:examples:2x2 [2018/05/22 14:21] (current)plewandowska Both sides previous revision Previous revision 2018/05/22 14:21 plewandowska 2018/05/22 14:21 plewandowska 2013/05/20 00:05 lpawela 2013/04/16 00:23 lpawela specialcases:2x2 renamed to numerical-shadow:examples:2x22013/03/20 16:25 lpawela 2013/03/20 16:25 lpawela 2013/03/20 16:21 lpawela 2013/03/20 15:37 lpawela 2013/03/14 13:32 lpawela 2013/03/13 12:39 lpawela 2013/03/13 12:31 lpawela 2013/03/13 12:31 lpawela 2013/03/13 12:28 lpawela 2013/03/13 12:23 lpawela created Next revision Previous revision 2018/05/22 14:21 plewandowska 2018/05/22 14:21 plewandowska 2013/05/20 00:05 lpawela 2013/04/16 00:23 lpawela specialcases:2x2 renamed to numerical-shadow:examples:2x22013/03/20 16:25 lpawela 2013/03/20 16:25 lpawela 2013/03/20 16:21 lpawela 2013/03/20 15:37 lpawela 2013/03/14 13:32 lpawela 2013/03/13 12:39 lpawela 2013/03/13 12:31 lpawela 2013/03/13 12:31 lpawela 2013/03/13 12:28 lpawela 2013/03/13 12:23 lpawela created Line 1: Line 1: + ~~NOTOC~~ ====== Hermitian matrices ====== ====== Hermitian matrices ====== In the case of Hermitian matrices the numerical shadow is a one dimensional distribution. In the case of Hermitian matrices the numerical shadow is a one dimensional distribution. Line 6: Line 7: <​html><​center>​ <​html><​center>​ - ^  Standard numerical shadow with respect to complex states. ​ ^ + ^  Standard numerical shadow with respect to complex ​states. ​ ^  Numerical shadow with respect to real states. ​ ^ - |  {{:​specialcases:​complex_a1.png?​nolink|Standard numerical shadow with respect to complex states}} ​ | + |  {{:​specialcases:​complex_a1.png?​nolink&300|Standard numerical shadow with respect to complex ​states}} ​ |  {{:​specialcases:​real_a1.png?​nolink&​300|Numerical shadow with respect to real states}} ​ | <​html>​ <​html>​ + + ====== Non-normal matrices ====== + + ==== Example 1 ==== + + The matrix is $$\begin{pmatrix} + 0&1\\ + 0&0 + \end{pmatrix}.$$ The shadow resembles a physical shadow cast by the hollow sphere made of transparent material when illuminated by a light source at infinity. + <​html><​center>​ <​html><​center>​ - ^  Standard numerical shadow with respect to real states. ​ ^ + ^  Standard numerical shadow with respect to complex ​states.  ^   ​Cross-section of the shadow along the real axis.  ^ - |  {{:specialcases:real_a1.png?​nolink|Numerical shadow with respect to real states}} ​ | + |  {{:numerical-shadow:​examples:​sec2fig1a.png?​nolink&​300|Standard numerical shadow with respect to complex states}} ​ |  {{:​numerical-shadow:​examples:sec2fig1b.png?nolink&300|Numerical shadow with respect to real states}} ​ | <​html>​ <​html>​ + ==== Example 2 ==== + The matrix is $$\begin{pmatrix} + 0&1\\ + 0&1 + \end{pmatrix}.$$ The shadow resembles physical shadow cast by the hollow sphere made of transparent material when illuminated by a light source at infinity, but with screen not perpendicular to the light rays. + <​html><​center>​ + ^  Standard numerical shadow with respect to complex states. ​ ^   ​Cross-section of the shadow along the real axis.  ^ + |  {{:​numerical-shadow:​examples:​sec2fig1c.png?​nolink&​300|Standard numerical shadow with respect to complex states}} ​ |  {{:​numerical-shadow:​examples:​sec2fig1d.png?​nolink&​300|Numerical shadow with respect to real states}} ​ | + <​html>