Random Matrix Theory

Bound the extreme eigenvalues of a sum of independent positive-semidefinite random matrices. The goal is the complete Matrix Chernoff theorem (Theorem 5.1.1): both expectation estimates, both tails, and the identities for the mean’s extreme eigenvalues.

The supporting milestone is the matrix mgf/cgf estimate (Lemma 5.4.1); Chapter 3 supplies the master bounds. The Lean statement retains the different lower- and upper-tail deviation ranges, permits zero mean eigenvalues, and handles a zero summand bound explicitly.

Source: Tropp, January 2015 version, Chapter 5. Lean statements compile; proofs remain open.

Shared definitions: Chapter 3; submit that draft first.

Series: Ch. 3 | Ch. 4 | Ch. 5 | Ch. 6 | Ch. 7 | Ch. 8.

Develop the reusable matrix Laplace-transform method behind Tropp’s concentration inequalities. The goal is Theorem 3.6.1: upper and lower tail and expectation bounds for sums of independent Hermitian random matrices, expressed through their matrix cumulant-generating functions.

Milestones cover the eigenvalue tail and expectation bounds (Propositions 3.2.1–2), probabilistic Lieb inequality (Corollary 3.4.2), and trace-cgf subadditivity (Lemma 3.5.1). This mission also supplies the series’ shared spectral, probability, and dilation definitions. Lieb’s underlying concavity theorem is the Chapter 8 goal.

The Lean statements give bounds for each admissible transform parameter, with explicit integrability hypotheses; optimization recovers the source’s infimum/supremum forms.

Source: Tropp, January 2015 version, Chapter 3. Lean statements compile; proofs remain open.

Series: Ch. 3 | Ch. 4 | Ch. 5 | Ch. 6 | Ch. 7 | Ch. 8.