# CSE 599I: The art and science of positive definite matrices

#### Instructor: James R. Lee

Office hours: By appointment

#### Teaching assistant:

• Ewin Tang (office hours by appointment)

#### Resources:

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 ] For a detailed exposition of matrix concentration: An Introduction to Matrix Concentration Inequalities Mon, Apr 05 Golden-Thompson and the Frobenius inner product [ scribe | video ] The Frobenius inner product and Schatten $p$-norms Interleaving correlations and the Golden-Thompson inequality The Golden-Thompson inequality — historical aspects and random matrix applications The Golden-Thompson inequality (Tao’s blog) Wed, Apr 07 Von Neumann's trace inequality and unitarily invariant norms [ scribe | video ] Section II (Majorization and Doubly Stochastic Matrices) in [BhatiaMA] Section IV (Symmetric Norms) in [BhatiaMA] Chapter 6 (Majorization and Singular Values) in [Hiai-Petz] Inequalities: Theory of Majorization and Its Applications, Marshall, Olkin, and Arnold Mon, Apr 12 Monotonicity and convexity [ scribe | video ] Operator monotonicity and convexity The Loewner-Heinz Theorem Wed, Apr 14 Joint convexity and the relative entropy [ scribe | video ] Mon, Apr 19 Lieb's concavity theorem [ scribe ] Wed, Apr 21 Quantum information theory [ scribe | video ] Supplementary reading: Chapters 1-3 in Petz’s book [QITPetz] Density matrices, measurements Pure states Composite systems and the partial trace Entanglement Quantum channels Failure of monotonicity for quantum entropy Mon, Apr 26 Quantum entropy and strong subadditivity [ scribe | video ] Summer school slides: Topics in Quantum Entropy and Entanglement (E. Lieb) Subadditivity and the failure of monotonicity Quantum mutual information and strong subadditivity Additional background on classical/quantum information and entropy: Chapters 10-11, in Wilde’s From Classical to Quantum Shannon Theory Wed, Apr 28 Nonncommutative averages [ scribe | video ] Quantum channels The quantum data processing inequality The operator Jensen inequality Jensen’s operator inequality (Hansen and Pedersen) Mon, May 03 Semidefinite programming and spectrahedra [ video ] Semidefinite programs Semidefinite characterizations of polytopes PSD rank Positive semidefinite rank (Fawzi, Gouveia, Parrilo, Robinson, Thomas) Extended formulations and lifts of polytopes Semidefinite programs and combinatorial optimization (Lovasz) Wed, May 05 The cut polytope and SOS cones [ video ] Mon, May 10 Symmetric cone factorizations [ scribe ] Wed, May 12 Sum of squares degree [ video ] Mon, May 17 Pattern matrices and smooth factorization [ scribe | video ] Wed, May 19 No lecture Mon, May 24 HW#3 discussion Homework #3 Wed, May 26 HW#3 discussion Wed, Jun 02 Quantum max-entropy approximation

## 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.