30 research outputs found
Numerical Ranges of KMS Matrices
A KMS matrix is one of the form J_n(a)=[{array}{ccccc} 0 & a & a^2 &... &
a^{n-1} & 0 & a & \ddots & \vdots & & \ddots & \ddots & a^2 & & & \ddots & a 0
& & & & 0{array}] for and in . Among other things,
we prove the following properties of its numerical range: (1) is a
circular disc if and only if and , (2) its boundary contains a line segment if and only if and , and (3)
the intersection of the boundaries and is either the singleton \{\min\sigma(\re J_n(a))\} if is
odd, and , or the empty set if otherwise, where,
for any -by- matrix , denotes its th principal submatrix
obtained by deleting its th row and th column (), \re A
its real part , and its spectrum.Comment: 35 page
Power Partial Isometry Index and Ascent of a Finite Matrix
We give a complete characterization of nonnegative integers and and a
positive integer for which there is an -by- matrix with its power
partial isometry index equal to and its ascent equal to . Recall that
the power partial isometry index of a matrix is the supremum,
possibly infinity, of nonnegative integers such that are all partial isometries while the ascent of is the smallest
integer for which equals . It was known
before that, for any matrix , either or
. In this paper, we prove more precisely that there is an
-by- matrix such that and if and only if one of the
following conditions holds: (a) , (b) and ,
and (c) and . This answers a question we asked in a previous
paper.Comment: 11 page
Weighted Shift Matrices: Unitary Equivalence, Reducibility and Numerical Ranges
An -by- () weighted shift matrix is one of the form
[{array}{cccc}0 & a_1 & & & 0 & \ddots & & & \ddots & a_{n-1} a_n & & &
0{array}], where the 's, called the weights of , are complex numbers.
Assume that all 's are nonzero and is an -by- weighted shift
matrix with weights . We show that is unitarily equivalent to
if and only if and, for some fixed , , () for all . Next, we show that
is reducible if and only if has periodic weights, that is, for some
fixed , , is divisible by , and
for all . Finally, we prove that and
have the same numerical range if and only if and
for all , where 's are the circularly symmetric functions.Comment: 27 page
Higher rank numerical ranges of normal matrices
The higher rank numerical range is closely connected to the construction of
quantum error correction code for a noisy quantum channel. It is known that if
a normal matrix has eigenvalues , then its higher
rank numerical range is the intersection of convex polygons with
vertices , where . In this paper, it is shown that the higher rank numerical range of a
normal matrix with distinct eigenvalues can be written as the intersection
of no more than closed half planes. In addition, given a convex
polygon a construction is given for a normal matrix
with minimum such that . In particular, if
has vertices, with , there is a normal matrix with such that .Comment: 12 pages, 9 figures, to appear in SIAM Journal on Matrix Analysis and
Application
Higher-rank Numerical Ranges and Kippenhahn Polynomials
We prove that two n-by-n matrices A and B have their rank-k numerical ranges
and equal to each other for all k, , if and only if their Kippenhahn polynomials
and coincide. The main tools for the proof are the Li-Sze
characterization of higher-rank numerical ranges, Weyl's perturbation theorem
for eigenvalues of Hermitian matrices and Bezout's theorem for the number of
common zeros for two homogeneous polynomials.Comment: 16 pages, 1 figur
Numerical ranges of reducible companion matrices
AbstractIn this paper, we show that a reducible companion matrix is completely determined by its numerical range, that is, if two reducible companion matrices have the same numerical range, then they must equal to each other. We also obtain a criterion for a reducible companion matrix to have an elliptic numerical range, put more precisely, we show that the numerical range of an n-by-n reducible companion matrix C is an elliptic disc if and only if C is unitarily equivalent to A⊕B, where A∈Mn-2, B∈M2 with σ(B)={aω1,aω2}, ω1n=ω2n=1, ω1≠ω2, and |a|⩾|ω1+ω2|+|ω1+ω2|2+4(1+2cos(π/n))/2