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

Spring, 2021

MW 3-4:20pm Remote on Zoom

Instructor: James R. Lee

Office hours: By appointment

Teaching assistant:

  • Ewin Tang (office hours by appointment)

Course email list [archives]


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.


Mon, Mar 29
PSD matrices
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  • 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
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Mon, Apr 05
Golden-Thompson and the Frobenius inner product
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Wed, Apr 07
Von Neumann's trace inequality and unitarily invariant norms
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Mon, Apr 12
Monotonicity and convexity
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  • Operator monotonicity and convexity
  • The Loewner-Heinz Theorem
Wed, Apr 14
Joint convexity and the relative entropy
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Mon, Apr 19
Lieb's concavity theorem
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Wed, Apr 21
Quantum information theory
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  • 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
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Wed, Apr 28
Nonncommutative averages
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Mon, May 03
Semidefinite programming and spectrahedra
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Wed, May 05
The cut polytope and SOS cones
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Mon, May 10
Symmetric cone factorizations
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Wed, May 12
Sum of squares degree
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Mon, May 17
Pattern matrices and smooth factorization
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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


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.