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Register a new userLow-dimensional reciprocal matrices with elliptical components of their Kippenhahn curves
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By definition, reciprocal matrices are tridiagonal $n$-by-$n$ matrices $A$ with constant main diagonal and such that $a_{i,i+1}a_{i+1,i}=1$ for $i=1,ldots,n-1$. For $nleq 6$, we establish criteria under which the numerical range generating curves (also called Kippenhahn curves) of such matrices consist of elliptical components only. As a corollary, we also provide a complete description of higher-rank numerical ranges when the criteria are met.
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Tridiagonal matrices with constant main diagonal and reciprocal pairs of off-diagonal entries are considered. Conditions for such matrices with sizes up to 6-by-6 to have elliptical numerical ranges are obtained.
The higher rank numerical ranges of generic matrices are described in terms of the components of their Kippenhahn curves. Cases of tridiagonal (in particular, reciprocal) 2-periodic matrices are treated in more detail.
A complete description of 4-by-4 matrices $begin{bmatrix}alpha I & C D & beta Iend{bmatrix}$, with scalar 2-by-2 diagonal blocks, for which the numerical range is the convex hull of two non-concentric ellipses is given. This result is obtained by reduction to the leading special case in which $C-D^*$ also is a scalar multiple of the identity. In particular cases when in addition $alpha-beta$ is real or pure imaginary, the results take an especially simple form. An application to reciprocal matrices is provided.
In this short paper, we review the Euler-Rodrigues formula for three-dimensional rotation via fractional powers of matrices. We derive the rotations by any angle through the spectral behavior of the fractional powers of the rotation matrix by $frac{pi}{2}$ in $mathbb{R}^3$ about some axis.
Non-reciprocal components, such as isolators and circulators, are critical to wireless communication and radar applications. Traditionally, non-reciprocal components have been implemented using ferrite materials, which exhibit non-reciprocity under the influence of an external magnetic field. However, ferrite materials cannot be integrated into IC fabrication processes, and consequently are bulky and expensive. In the recent past, there has been strong interest in achieving non-reciprocity in a non-magnetic IC-compatible fashion using spatio-temporal modulation. In this paper, we present a general approach to non-reciprocity based on switched transmission lines. Switched transmission lines enable broadband, lossless and compact non-reciprocity, and a wide range of non-reciprocal functionalities, including non-reciprocal phase shifters, ultra-broadband gyrators and isolators, frequency-conversion isolators, and high-linearity/high-frequency/ultra-broadband circulators. We present a detailed theoretical analysis of the various non-idealities that impact insertion loss and provide design guidelines. The theory is validated by experimental results from discrete-component-based gyrators and isolators, and a 25GHz circulator fabricated in 45nm SOI CMOS technology.
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