The nonzero eigenvalues of $AB$ are equal to those of $BA$: an identity that
holds as long as the products are square, even when $A,B$ are rectangular. This
fact naturally suggests an efficient algorithm for computing eigenvalues and
eigenvectors of a low-rank matrix $X= AB$ with $A,B^T\in\mathbb{C}^{N\times r},
N\gg r$: form the small $r\times r$ matrix $BA$ and find its eigenvalues and
eigenvectors. For nonzero eigenvalues, the eigenvectors are related by $ ABv =
\lambda v \Leftrightarrow BAw = \lambda w $ with $w=Bv$, and the same holds for
Jordan vectors. For zero eigenvalues, the Jordan blocks can change sizes
between $AB$ and $BA$, and we characterize this behavior.

矩阵乘积的非零特征值相等定理在矩阵计算的应用中有很大的优势，因为它可以用于计算低秩矩阵 AB 的特征值和特征向量，即求出矩阵 BA 的特征值和特征向量，从而得到原矩阵 AB 的特征值和特征向量。此外，本文还讨论了 Jordan 块在 AB 和 BA 之间的差异。

低秩特征值问题

The low-rank eigenvalue problem

In the past few years, successive improvements of the asymptotic complexity
of square matrix multiplication have been obtained by developing novel methods
to analyze the powers of the Coppersmith-Winograd tensor, a basic construction
introduced thirty years ago. In this paper we show how to generalize this
approach to make progress on the complexity of rectangular matrix
multiplication as well, by developing a framework to analyze powers of tensors
in an asymmetric way. By applying this methodology to the fourth power of the
Coppersmith-Winograd tensor, we succeed in improving the complexity of
rectangular matrix multiplication. Let $\alpha$ denote the maximum value such
that the product of an $n\times n^\alpha$ matrix by an $n^\alpha\times n$
matrix can be computed with $O(n^{2+\epsilon})$ arithmetic operations for any
$\epsilon>0$. By analyzing the fourth power of the Coppersmith-Winograd tensor
using our methods, we obtain the new lower bound $\alpha>0.31389$, which
improves the previous lower bound $\alpha>0.30298$ obtained five years ago by
Le Gall (FOCS'12) from the analysis of the second power of the
Coppersmith-Winograd tensor. More generally, we give faster algorithms
computing the product of an $n\times n^k$ matrix by an $n^k\times n$ matrix for
any value $k\neq 1$. (In the case $k=1$, we recover the bounds recently
obtained for square matrix multiplication). These improvements immediately lead
to improvements in the complexity of a multitude of fundamental problems for
which the bottleneck is rectangular matrix multiplication, such as computing
the all-pair shortest paths in directed graphs with bounded weights.

本文提出一种利用张量幂次的对称和非对称结构实现矩阵乘法复杂度下界优化的新方法，通过分析 Coppersmith-Winograd 张量的四次幂进一步降低矩阵乘法复杂度下界，并得到了许多基于矩阵乘法的问题的新解法。

利用 Coppersmith-Winograd 张量的幂次改进矩阵乘法

Improved Rectangular Matrix Multiplication using Powers of the  Coppersmith-Winograd Tensor

We present a convex formulation of dictionary learning for sparse signal
decomposition. Convexity is obtained by replacing the usual explicit upper
bound on the dictionary size by a convex rank-reducing term similar to the
trace norm. In particular, our formulation introduces an explicit trade-off
between size and sparsity of the decomposition of rectangular matrices. Using a
large set of synthetic examples, we compare the estimation abilities of the
convex and non-convex approaches, showing that while the convex formulation has
a single local minimum, this may lead in some cases to performance which is
inferior to the local minima of the non-convex formulation.

本文介绍了用于稀疏信号分解的词典学习的凸优化方法，通过引入凸排名降低项替代字典大小的上限来获得凸性。在一系列的合成示例中，分类比较了凸优化方法和非凸优化方法的估计能力，结果表明，虽然凸优化方法只有一个局部最小值，但在某些情况下，其性能可能不如非凸优化方法的局部最小值。