635 research outputs found

### Positivity of Partitioned Hermitian Matrices with Unitarily Invariant Norms

We give a short proof of a recent result of Drury on the positivity of a
$3\times 3$ matrix of the form $(\|R_i^*R_j\|_{\rm tr})_{1 \le i, j \le 3}$ for
any rectangular complex (or real) matrices $R_1, R_2, R_3$ so that the
multiplication $R_i^*R_j$ is compatible for all $i, j$, where $\|\cdot\|_{\rm
tr}$ denotes the trace norm. We then give a complete analysis of the problem
when the trace norm is replaced by other unitarily invariant norms.Comment: 6 page

### Determinantal and eigenvalue inequalities for matrices with numerical ranges in a sector

Let A = \pmatrix A_{11} & A_{12} \cr A_{21} & A_{22}\cr\pmatrix \in M_n,
where $A_{11} \in M_m$ with $m \le n/2$, be such that the numerical range of
$A$ lies in the set \{e^{i\varphi} z \in \IC: |\Im z| \le (\Re z) \tan
\alpha\}, for some $\varphi \in [0, 2\pi)$ and $\alpha \in [0, \pi/2)$. We
obtain the optimal containment region for the generalized eigenvalue $\lambda$
satisfying \lambda \pmatrix A_{11} & 0 \cr 0 & A_{22}\cr\pmatrix x = \pmatrix
0 & A_{12} \cr A_{21} & 0\cr\pmatrix x \quad \hbox{for some nonzero} x \in
\IC^n, and the optimal eigenvalue containment region of the matrix $I_m -
A_{11}^{-1}A_{12} A_{22}^{-1}A_{21}$ in case $A_{11}$ and $A_{22}$ are
invertible. From this result, one can show $|\det(A)| \le \sec^{2m}(\alpha)
|\det(A_{11})\det(A_{22})|$. In particular, if $A$ is a accretive-dissipative
matrix, then $|\det(A)| \le 2^m |\det(A_{11})\det(A_{22})|$. These affirm some
conjectures of Drury and Lin.Comment: 6 pages, to appear in Journal of Mathematical Analysi

### Canonical forms, higher rank numerical range, convexity, totally isotropic subspace, matrix equations

Results on matrix canonical forms are used to give a complete description of
the higher rank numerical range of matrices arising from the study of quantum
error correction. It is shown that the set can be obtained as the intersection
of closed half planes (of complex numbers). As a result, it is always a convex
set in $\mathcal C$. Moreover, the higher rank numerical range of a normal
matrix is a convex polygon determined by the eigenvalues. These two
consequences confirm the conjectures of Choi et al. on the subject. In
addition, the results are used to derive a formula for the optimal upper bound
for the dimension of a totally isotropic subspace of a square matrix, and
verify the solvability of certain matrix equations.Comment: 10 pages. To appear in Proceedings of the American Mathematical
Societ

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