Linear algebra and group theory

Abstract

This is an introduction to linear algebra and group theory. We first review the linear algebra basics, namely the determinant, the diagonalization procedure and more, and with the determinant being constructed as it should, as a signed volume. We discuss then the basic applications of linear algebra to questions in analysis. Then we get into the study of the closed groups of unitary matrices $G\subset U_N$, with some basic algebraic theory, and with a number of probability computations, in the finite group case. In the general case, where $G\subset U_N$ is compact, we explain how the Weingarten integration formula works, and we present some basic $N\to \infty$ applications.

400 pages

References in corpus (9)

• Moments and Cumulants of Polynomial random variables on unitary groups, the Itzykson-Zuber integral and free probability
• On some properties of orthogonal Weingarten functions
• Random quantum channels I: graphical calculus and the Bell state phenomenon
• The hyperoctahedral quantum group
• Integration over quantum permutation groups
• Sinkhorn normal form for unitary matrices
• Gaussianization and eigenvalue statistics for random quantum channels (III)
• On polynomial integrals over the orthogonal group
• Woronowicz's Tannaka-Krein duality and free orthogonal quantum groups