The art and science of positive definite matrices

Spring, 2021

MW 3-4:20pm Remote on Zoom

Instructor: James R. Lee

Office hours: TBD

Teaching assistant:

  • Ewin Tang

Course email list [archives]

Homeworks

Course description:

Positive semidefinite matrices are fundamental objects in semidefinite programming, quantum information theory, and spectral graph theory. Despite their widespread utility, analysis and geometry on the PSD cone is often strange, subtle and, occasionally, magical. This course will focus on the properties of such matrices with an eye toward applications.

Often this gives certain phenomena an "operational" interpretation that provides intuition to complement the underlying linear algebra. The style of the course will be to first introduce a classical argument with real numbers and then to explore analogs for PSD matrices.

Lectures

Mon, Mar 29
PSD matrices
[ scribe | video ]
  • Elementary facts about Hermitian matrices
  • PSD matrices are Gram matrices
  • The Loewner order
  • The Hadamard (aka Schur) product of matrices as a principal submatrix of the tensor product

  • [BhatiaPDM] Sec 1.1-1.2
Wed, Mar 31
Sum of independent random matrices
[ scribe | video ]
Mon, Apr 05
Golden-Thompson and the Frobenius inner product
[ scribe | video ]
Wed, Apr 07
Von Neumann's trace inequality and unitarily invariant norms
[ scribe | video ]
Mon, Apr 12
Operator monotonicity and convexity

Homeworks

You may discuss problems with your classmates, but when you write down the solutions, you should do so by yourself. You can use the internet and books for reference material but you should cite every source that you consulted (the name of the book or web page suffices). You should also cite any classmates with whom you discussed solutions. Homework should be typeset.