Diagonal imbeddings in a normal matrix
Introduction
The numerical range of a matrix is a closed, convex subset of the complex plane defined byand when is normal, coincides with the convex hull of its eigenvalues, i.e. . This set has attracted a lot of interest, especially in the context of numerical solutions of partial differential equations, stability analysis of dynamical systems and convergence theory of matrix iterations, among other applications. For an introduction to this area, see [7], [8].
Given , a unit vector is a generating vector for if . Recently in [15], for a given matrix A and a point the inverse problem of finding a generating vector x for was studied. This is a problem relating the image of the quadratic mapto its domain, the complex unit sphere and it has been proved in [2] that there exist n linearly independent generating vectors for any such point. Algorithms computing generating vectors can be found in [15], [2].
For the pair of matrices and (), we say that B is imbeddable in A if there exists an isometry (i.e. ) such that . The problem of imbedding for Hermitian matrices has been investigated in [6], while several necessary imbedding conditions for a normal matrix B to be imbeddable in normal A and related results have been presented in [5], [14], [9], [10], [11]. In the special case , this problem has been studied via higher rank numerical ranges [3],which have recently found applications in quantum error correction.
In this paper, we propose a new and more general inverse numerical range problem, involving the construction of subspaces of dimension greater than one that yield diagonal matrices imbeddable in A. For a normal matrix and a point , we seek to determine the minimum integer , such that the relationship is satisfied, for an isometry and suitable, not unique points to be determined. In particular, for the smallest k we construct mutually orthonormal and A-orthogonal vectors (i.e. , for ) that define the isometry . Then, is a matrix of maximum order imbeddable in A and W will be referred to as generating isometry for the diagonal B in the normal A. It should be pointed out that nothing relevant to this problem has been presented in the literature. Our results may be helpful for the construction of generating isometries corresponding to points in higher rank numerical ranges. This is crucial for the construction of error correcting codes in quantum computing and remains an open problem in this area [4].
Our paper is organized as follows: in Section 2 the smallest integer k for which a diagonal matrix should be imbeddable in A is determined and a counterexample shows that this bound is the best possible. Further in Section 3, a recursive procedure to produce mutually orthogonal and A-orthogonal unit vectors is proposed and consequently an isometry is constructed, such that the matrix is diagonal. Moreover, some useful properties, derived from diagonal imbeddability, are presented. In the last section, we conclude our paper presenting a generalization of a result in [1], which implies a method to obtain generating vectors for points and to construct chains of orthonormal bases for subspaces in which the columns of W recursively lie.
Section snippets
Maximality of a diagonal matrix imbedded in a normal matrix
Let a normal matrix and a point that is not an eigenvalue. We consider in this section the problem to find the minimum integer and an isometry with the property that for suitable points .
In the case the eigenvalues of A are collinear, the polygon degenerates to a line segment. If denotes the line on which the eigenvalues of A lie, is the slope of and , then the matrix A is equal to ,
Construction of mutually orthogonal and A-orthogonal vectors
As a first step we present a method to obtain generating vectors for a given point , when is a non-degenerate polygon. We remind that are the orthonormal eigenvectors of the normal matrix A corresponding to its eigenvalues . Hence, due to convexity we have:and then clearly a generating vector for isFor , the expression (4) is not unique and the coefficients may be chosen to be nonzero.
For
A geometric procedure
In this last section we structure an orthonormal basis for via an approach, which gives a geometric aspect of the problem. We begin with a preliminary result in the next Proposition, which is an extension of Proposition 1 in [1] and implies a procedure to obtain recursive sequences of biorthogonal vectors. Proposition 4 Let be a normal matrix with unit eigenvectors corresponding to its eigenvalues and let the sequences of unit vectors and of defined recursively by
References (15)
- et al.
On compressions of normal matrices
Linear Algebra Appl.
(2002) - et al.
Higher-rank numerical ranges and compression problems
Linear Algebra Appl.
(2006) - et al.
On normal extensions of submatrices
Linear Algebra Appl.
(2003) - et al.
Imbedding conditions for normal matrices
Linear Algebra Appl.
(2009) A simple algorithm for the inverse field of values problem
Inverse Problems
(2009)- et al.
Higher-rank numerical ranges of unitary and normal matrices
Operators and Matrices
(2007) - et al.
Generalized minimax and interlacing theorems
Linear Multilinear Algebra
(1984)
Cited by (4)
Interaction between Hermitian and normal imbeddings
2014, Applied Mathematics and ComputationThe imbeddability for hermitian and normal matrices
2013, Linear Algebra and Its ApplicationsBlock imbedding and interlacing results for normal matrices
2016, Electronic Journal of Linear AlgebraAn inverse problem for the κ-rank numerical range
2016, SIAM Journal on Matrix Analysis and Applications