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numerical-range:generalizations:higher-rank-numerical-range [2018/11/03 22:54]
lpawela
numerical-range:generalizations:higher-rank-numerical-range [2020/04/06 15:15] (current)
rkukulski
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 The rank-$k$ The rank-$k$
-numerical ranges, denoted below by $\Lambda_k$,​ were introduced ​c. 2006 by Choi, Kribs, and Życzkowski +numerical ranges, denoted below by $\Lambda_k$,​ were introduced by Choi, Kribs, and Życzkowski 
-as a tool to handle compression problems in quantum information theory. Since then +as a tool to handle compression problems in quantum information theory ​(See for details [[numerical-range:​generalizations:​application-of-higher-rank-and-p-k-numerical-range|Application of higher rank numerical range]]). Since then 
-their theory and applications have been advanced with remarkable enthusiasm. ​The +their theory and applications have been advanced with remarkable enthusiasm.  
-sequence of papers ​[:​choi2007higher ​)], [( :choi2008geometry ​)], [( :​woerdeman2008higher )], [( :li2008canonical ​)], for example, led to a striking +Among the many more recent ​papers ​concerning the $\Lambda_k(M)$let us mention ​  [( :choi2007higher ​)], [( :​woerdeman2008higher )], [( :li2009condition ​)], [( :gau2010higher ​)], [( :chien2020diameter ​)] and [( :​holbrook2015bounds )].
-extension of the classical Toeplitz--Hausdorff theorem (convexity of $W(M)$): **all** the +
-$\Lambda_k(M)$ are convex (though some may be empty), and they are intersections +
-of conveniently computable half-planes in $\mathbb{C}$. +
-Among the many more recent papers concerning the $\Lambda_k(M)$,​ let us mention ​[( :li2009condition ​)], [( :gau2010higher ​)] and [( :​holbrook2015bounds )].+
  
-Given a matrix $M$ of dimension $d$ and $k\geq1$, Choi, Kribs, and Życzkowski (see [( :​choi2006higher )])+Given a matrix $M$ of dimension $d$ and $k\geq1$, Choi, Kribs, and Życzkowski (see [( :​choi2006higher ​)], [( :​choi2008geometry ​)])
 defined the rank-$k$ numerical range of $M$ as defined the rank-$k$ numerical range of $M$ as
 \[ \[
 \Lambda_k(M)=\{\lambda\in\mathbb{C}:​\exists P\in P_k\mbox{ such that }PMP=\lambda P\}, \Lambda_k(M)=\{\lambda\in\mathbb{C}:​\exists P\in P_k\mbox{ such that }PMP=\lambda P\},
 \] \]
-where $P_k$ denotes the set of rank-$k$ orthogonal projections in $M_d$. It is not hard+where $P_k$ denotes the set of rank-$k$ orthogonal projections in $M_d$
 +This definition is natural extention to [[:​numerical-range|classical numerical range]]It is easy to see that if we considek $k=1$, then $ \Lambda_1(A) = W(A)$ for any matrix $A \in M_d$.  
 +==== Alternative definitions ==== 
 + 
 +Let $M \in M_d$. Then higher rank-$k$ numerical range of $M$ we can alternatively define as  
 +\begin{equation} 
 +\Lambda_k(M) = \left\{ \alpha \in \mathbb{C}: U^\dagger M U = \left[\begin{array}{cc}\alpha \1_k&* \\ * &​*\end{array}\right] \text{for some unitary } U \right\}. 
 +\end{equation}  
 + 
 +Equivalently ​ [( :​li2008canonical )],  
 + 
 +\begin{equation} 
 +\Lambda_k(M) = \left\{\alpha \in \mathbb{C}: e^{i \xi}\alpha + e^{-i\xi} \bar{\alpha} \le  \lambda_k \left( e^{i \xi}M + e^{-i\xi} M^\dagger ​ \right) \text{for all } \xi \in [0,2\pi) \right\} 
 +\end{equation} 
 + where $\lambda_k(X)$ is $k$-th eigenvalue of given matrix $X$.  
 +  
 +===== Properties ===== 
 +If $M$ is a normal matrix with eigenvalues $m_1, \ldots, m_n$, then  
 +\begin{equation} 
 +\Lambda_k(M) = \bigcap_{1 \le j_1 < \ldots < j_{n-k+1} \le n } \mathrm{conv} \left\{ m_{j_1}, \ldots , m_{j_{n-k+1}}\right\}. 
 +\end{equation}  
 + It is not hard
 to verify that $\Lambda_K(M)$ can also be described as the set of complex $\lambda$ such that to verify that $\Lambda_K(M)$ can also be described as the set of complex $\lambda$ such that
 there is some $k$-dimensional ​ subspace $S$ of $\mathbb{C}^d$ such that $(Mu,​u)=\lambda$ for **all** there is some $k$-dimensional ​ subspace $S$ of $\mathbb{C}^d$ such that $(Mu,​u)=\lambda$ for **all**
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 \] \]
  
-===== Examples ​===== +===== Special cases ===== 
-== Hermitian matrices ​==+For matrices of a block diagonal form: 
 +\[ 
 +J_n(\alpha)\oplus \beta \1_m, 
 +\]  
 +where $J_n(\alpha)$ is a Jordan matrix with eigenvalue $\alpha$, the full characterization of rank-$k$ numerical range was studied in [( :​argerami2019higher )]. 
 + 
 +=== Comparison between $k$-numerical range and higher-rank numerical range ===
 A comparison between the [[:​numerical-range:​generalizations:​k-numerical-range|$k$-numerical range]] and higher-rank numerical range in the case $k=2$. Note that $\Lambda_2 \subset W_2$. The matrix used in this example is $A = \mathrm{diag}(1,​ 2, 4, 8)$. A comparison between the [[:​numerical-range:​generalizations:​k-numerical-range|$k$-numerical range]] and higher-rank numerical range in the case $k=2$. Note that $\Lambda_2 \subset W_2$. The matrix used in this example is $A = \mathrm{diag}(1,​ 2, 4, 8)$.
 {{ :​numerical-range:​examples:​k-range.png?​nolink&​500 |}} {{ :​numerical-range:​examples:​k-range.png?​nolink&​500 |}}
  
-== Unitary matrices == +===== Examples ===== 
-Numerical range (blue) and real numerical shadow of the matrix $U_5 = \mathrm{diag}(\left\{\ee^{2\pi \ii k/​5}\right\}_{k=1}^5)$. The red polygon inside is $\Lambda_2(U_5)$.+Undermentioned examples are made by Raymond Nung-Sing Sze. 
 +=== Unitary matrices ​=== 
 +1. Consider a diagonal unitary ​matrix $U_5 = \mathrm{diag}(\left\{\ee^{2\pi \ii k/​5}\right\}_{k=1}^5)$. The blue line represents the numerical range $\Lambda_1(U_5) = W(U_5)$ and the grey figure the real numerical shadow of $U_5$. The red polygon inside is $\Lambda_2(U_5)$.
 {{ :​numerical-range:​generalizations:​real_s3d5.png?​nolink&​500 |}} {{ :​numerical-range:​generalizations:​real_s3d5.png?​nolink&​500 |}}
  
-Numerical range (blue) and real numerical shadow of the matrix $U_7 = \mathrm{diag}(\left\{\ee^{2\pi \ii k/5}\right\}_{k=1}^7)$. The red polygon inside is $\Lambda_2(U_7)$ and the green polygon is $\Lambda_3(U_7)$.+2.  Consider a diagonal unitary ​matrix ​ $U_7 = \mathrm{diag}(\left\{\ee^{2\pi \ii k/7}\right\}_{k=1}^7)$. The blue line represents the numerical range $\Lambda_1(U_7) = W(U_7)$ and the grey figure the real numerical shadow of $U_7$. The red polygon inside is higher $2$-rank numerical range $\Lambda_2(U_7)$ and the green polygon is higher $3$-rank numerical range $\Lambda_3(U_7)$.
 {{ :​numerical-range:​generalizations:​real_s3d7.png?​nolink&​500 |}} {{ :​numerical-range:​generalizations:​real_s3d7.png?​nolink&​500 |}}
 +
 +3. Consider a diagonal unitary matrix ​ $A= \mathrm{diag}(\left\{\ee^{2\pi \ii k/​9}\right\}_{k=1}^9)$. The first picture represents $\Lambda_1(A) = W(A) $ classical numerical range of $A$ whereas the second picture represents higher $2$-rank numerical range $\Lambda_2(A)$.
 +{{ :​numerical-range:​generalizations:​normal003.png |}}
 +{{ :​numerical-range:​generalizations:​normal004.png |}}
 +
 +=== Non-normal matrices === 
 +Let $$A=\begin{pmatrix}
 +\ii & 0 & 0 & 0 & 0 \\
 +0 & -\ii & 0 & 0 & 0 \\
 +0 & 0 & -1 & 0 & 0  \\
 +0 & 0 & 0 & -2 & 1 \\
 +0 & 0 & 0 & 0 & 0
 +\end{pmatrix}$$
 +will be non-normal matrix. Its numerical range is given by the first picture and higher $2$-rank numerical range is presented on the second picture. ​
 +{{ :​numerical-range:​generalizations:​example105.png |}}
 +{{ :​numerical-range:​generalizations:​example106.png |}}
 +
 +
 +
numerical-range/generalizations/higher-rank-numerical-range.1541285670.txt.gz · Last modified: 2018/11/03 22:54 by lpawela