Differences

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

--- 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
@@ Line 4: / Line 4: @@
 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
 \[
 \Lambda_k(M)=\{\lambda\in\mathbb{C}:\exists P\in P_k\mbox{ such that }PMP=\lambda P\},
 \]
-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
 there is some $k$-dimensional  subspace $S$ of $\mathbb{C}^d$ such that $(Mu,u)=\lambda$ for **all**
@@ Line 39: / Line 56: @@
 \]
-===== 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 $A = \mathrm{diag}(1, 2, 4, 8)$.
 {{ :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 ===
+. 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 (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 $\Lambda_3(U_7)$.
+.  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 |}}
+. 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 $$A=\begin{pmatrix}
+\ii & 0 & 0 & 0 & 0 \\
+& -\ii & 0 & 0 & 0 \\
+& 0 & -1 & 0 & 0  \\
+& 0 & 0 & -2 & 1 \\
+& 0 & 0 & 0 & 0
+\end{pmatrix}$$
+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 Shadow

User Tools

Site Tools

Differences

Page Tools