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NUMERICAL RANGE AND POSITIVE BLOCK MATRICES
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 103 / Issue 1 / February 2021
- Published online by Cambridge University Press:
- 11 June 2020, pp. 69-77
- Print publication:
- February 2021
-
- Article
- Export citation
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We obtain several norm and eigenvalue inequalities for positive matrices partitioned into four blocks. The results involve the numerical range
$W(X)$ of the off-diagonal block 
$X$, especially the distance 
$d$ from 
$0$ to 
$W(X)$. A special consequence is an estimate, between the diameters of the numerical ranges for the full matrix and its partial trace.
$$\begin{eqnarray}\text{diam}\,W\left(\left[\begin{array}{@{}cc@{}}A & X\\ X^{\ast } & B\end{array}\right]\right)-\text{diam}\,W\biggl(\frac{A+B}{2}\biggr)\geq 2d,\end{eqnarray}$$
