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numerical-range:generalizations:q-numerical-range

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numerical-range:generalizations:q-numerical-range [2019/06/11 14:31]
plewandowska [Definition]
numerical-range:generalizations:q-numerical-range [2019/10/01 12:36] (current)
plewandowska
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 ===== Properties ===== ===== Properties =====
-Properties of $W_q(A)$ of a matrix $A$ of dimension $d \times d$ [( :chien2006q )],[( :​chien2002davis )]:+Properties of $W_q(A)$ of a matrix $A$ of dimension $d \times d$ [( :li1998q )], [( :chien2006q )],[( :​chien2002davis )]:
  
    - Note that in the case $q=1$ we get back the [[:​numerical-range|standard numerical range]];    - Note that in the case $q=1$ we get back the [[:​numerical-range|standard numerical range]];
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    - For any $\mu, \eta \in \mathbb{C}$,​ we have $W_q(\mu A + \eta \1) = \mu W_q(A) + \eta q$.    - For any $\mu, \eta \in \mathbb{C}$,​ we have $W_q(\mu A + \eta \1) = \mu W_q(A) + \eta q$.
  
 +=====Example===== ​
 +The numerical aproximation of $W_{13/​14}(A)$,​ where $$
 +A = \begin{pmatrix}
 +0&​1&​1/​2\\
 +0&​0&​1\\
 +0&​0&​0
 +\end{pmatrix}.
 +$$
 +
 +
 +{{:​numerical-range:​qnr1.png?​nolink|}}  ​
 +
 +
 +This example can be found in [( :koide2008q )]. 
numerical-range/generalizations/q-numerical-range.1560263516.txt.gz · Last modified: 2019/06/11 14:31 by plewandowska