Numerical Shadow

The web resource on numerical range and numerical shadow

User Tools

Site Tools


numerical-range:generalizations:q-numerical-range

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision
Previous revision
Last revision Both sides next revision
numerical-range:generalizations:q-numerical-range [2019/06/11 14:31]
plewandowska [q--numerical range]
numerical-range:generalizations:q-numerical-range [2019/10/01 12:36]
plewandowska
Line 5: Line 5:
 Let $A$ be an $d \times d$ matrix. The q-numerical range of $A$ is the set: Let $A$ be an $d \times d$ matrix. The q-numerical range of $A$ is the set:
 \[ \[
-W_q(A) = \left\{ z \in \mathbb{C}: z = \bra{x}A\ket{y}, ​|\braket{x}{y}= q, \ket{x} \in \mathbb{C}^d,​ \ket{y} \in \mathbb{C}^d,​ \braket{x}{x} = 1, \braket{y}{y} = 1 \right\}.+W_q(A) = \left\{ z \in \mathbb{C}: z = \bra{x}A\ket{y},​ \braket{x}{y} = q, \ket{x} \in \mathbb{C}^d,​ \ket{y} \in \mathbb{C}^d,​ \braket{x}{x} = 1, \braket{y}{y} = 1 \right\}.
 \] \]
  
 ===== 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]];
Line 18: Line 18:
    - 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&​525}}  ​
 +
 +
 +This example can be found in [( :koide2008q )]. 
numerical-range/generalizations/q-numerical-range.txt · Last modified: 2019/10/01 12:36 by plewandowska