Weighted Shift Matrices: Unitary Equivalence, Reducibility and Numerical Ranges

An $n$-by-$n$ ($n\ge 3$) weighted shift matrix $A$ is one of the form [{array}{cccc}0 & a_1 & & & 0 & \ddots & & & \ddots & a_{n-1} a_n & & & 0{array}], where the $a_j$'s, called the weights of $A$, are complex numbers. Assume that all $a_j$'s are nonzero and $B$ is an $n$-by-$n$ weighted shift matrix with weights $b_1,..., b_n$. We show that $B$ is unitarily equivalent to $A$ if and only if $b_1... b_n=a_1...a_n$ and, for some fixed $k$, $1\le k \le n$, $|b_j| = |a_{k+j}|$ ($a_{n+j}\equiv a_j$) for all $j$. Next, we show that $A$ is reducible if and only if $A$ has periodic weights, that is, for some fixed $k$, $1\le k \le \lfloor n/2\rfloor$, $n$ is divisible by $k$, and $|a_j|=|a_{k+j}|$ for all $1\le j\le n-k$. Finally, we prove that $A$ and $B$ have the same numerical range if and only if $a_1...a_n=b_1...b_n$ and $S_r(|a_1|^2,..., |a_n|^2)=S_r(|b_1|^2,..., |b_n|^2)$ for all $1\le r\le \lfloor n/2\rfloor$, where $S_r$'s are the circularly symmetric functions.Comment: 27 page

SegFlow: Joint Learning for Video Object Segmentation and Optical Flow

This paper proposes an end-to-end trainable network, SegFlow, for simultaneously predicting pixel-wise object segmentation and optical flow in videos. The proposed SegFlow has two branches where useful information of object segmentation and optical flow is propagated bidirectionally in a unified framework. The segmentation branch is based on a fully convolutional network, which has been proved effective in image segmentation task, and the optical flow branch takes advantage of the FlowNet model. The unified framework is trained iteratively offline to learn a generic notion, and fine-tuned online for specific objects. Extensive experiments on both the video object segmentation and optical flow datasets demonstrate that introducing optical flow improves the performance of segmentation and vice versa, against the state-of-the-art algorithms.Comment: Accepted in ICCV'17. Code is available at https://sites.google.com/site/yihsuantsai/research/iccv17-segflo