A linear matrix is a matrix whose entries are linear forms in some indeterminates $t_1,\dots, t_m$ with coefficients in some field $F$. The *commutative rank* of a linear matrix is obtained by interpreting it as a matrix with entries in the function field $F(t_1,\dots,t_m)$, and is directly related to the central PIT (polynomial identity testing) problem. The*non-commutative rank* of a linear matrix is obtained by interpreting it as a matrix with entries over the free skew field. The non-commutative rank could be larger than the commutative rank. Many incarnations and applications of non-commutative rank across areas of math, physics and CS have been found, some recently, and there is scope for more.

In this talk, I will discuss several interesting structural and algorithmic aspects of non-commutative rank, as well as the role it plays in non-commutative rational identity testing and tensor rank lower bounds. Time permitting, I will indicate the connections to invariant theory for quivers, operator scaling and Brascamp--Lieb inequalities.

No special background beyond standard linear algebra will be assumed.

# Non-commutative rank

Visu Makam

University of Michigan; Member, School of Mathematics

February 5, 2019