# Numerical Shadow

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# Higher rank numerical range

## Definition

The rank-$k$ numerical ranges, denoted below by $\Lambda_k$, were introduced c. 2006 by Choi, Kribs, and Życzkowski as a tool to handle compression problems in quantum information theory. Since then their theory and applications have been advanced with remarkable enthusiasm. The sequence of papers , , , , for example, led to a striking 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 ,  and .

Given a matrix $M$ of dimension $d$ and $k\geq1$, Choi, Kribs, and Życzkowski (see ) 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 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 unit vectors in $S$. In particular, we see that $W(M)=\Lambda_1(M)\supseteq\Lambda_2(M)\supseteq\Lambda_3(M)\supseteq\dots\quad.$

Note that, this numerical range is different from the $k$-numerical range as for a Hermitian matrix $A$, we get $\Lambda_k(A) = [\lambda_k, \lambda_{N-k+1}],$ where $\lambda_k$ are the eigenvalues of $A$ in an increasing order. On the other hand, the $k$-numerical range is given by $W_k = \left[\frac{1}{k}\sum_{i=1}^k\lambda_i, \frac{1}{k}\sum_{i=0}^{k-1} \lambda_{d-i} \right].$ Hence, we get $\Lambda_k(A) \subset W_k(A).$

## Examples

##### Hermitian matrices

A comparison between the $k$-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 $A = \mathrm{diag}(1, 2, 4, 8)$. ##### Unitary matrices  1. M. D. Choi, J. A. Holbrook, D. W, Kribs, K. Życzkowski, 2007. Higher-rank numerical ranges of unitary and normal matrices. Operators and Matrices, 1, pp.409–426.
2. M. D. Choi, M. Giesinger, J. A. Holbrook, D. W. Kribs, 2008. Geometry of higher-rank numerical ranges. Linear and Multilinear Algebra, 56, Taylor & Francis, pp.53–64.
3. H. J. Woerdeman, 2008. The higher rank numerical range is convex. Linear and Multilinear Algebra, 56, Taylor & Francis, pp.65–67.
4. C. K, Li, N. S. Sze, 2008. Canonical forms, higher rank numerical ranges, totally isotropic subspaces, and matrix equations. Proceedings of the American Mathematical Society, 136, pp.3013–3023.
5. C. K. Li, Y. T. Poon, N. S. Sze, 2009. Condition for the higher rank numerical range to be non-empty. Linear and Multilinear Algebra, 57, Taylor & Francis, pp.365–368.
6. Hwa-Long Gau, Chi-Kwong Li, Pei Yuan Wu, 2010. Higher-rank numerical ranges and dilations. Journal of Operator Theory, JSTOR, pp.181–189.
7. John Holbrook, Nishan Mudalige, Mike Newman, Rajesh Pereira, 2015. Bounds on polygons of higher rank numerical ranges. Linear Algebra and its Applications, 474, Elsevier, pp.230–242.
8. M. D. Choi, D. W. Kribs, K. Życzkowski, 2006. Higher-rank numerical ranges and compression problems. Linear algebra and its applications, 418, pp.828–839. 