Diagonalization
Stub. Diagonalizability conditions, upper-triangular matrices as a stepping stone, the Gershgorin disk theorem for locating eigenvalues, commuting operators, and applications — matrix limits/Markov chains and the Cayley-Hamilton theorem.
When can an operator be represented by a diagonal matrix? Builds up via upper-triangular representability (always possible over ), states the exact diagonalizability conditions, and covers the tools that come with the territory: locating eigenvalues without computing them (Gershgorin disks), simultaneous diagonalizability of commuting operators, and two payoff applications — long-run behavior of Markov chains via matrix limits, and the Cayley–Hamilton theorem (every operator satisfies its own characteristic polynomial), which anticipates the invariant-subspace decomposition of the canonical-forms note.
Outline (TODO — flesh out each)
- Upper-triangular matrices
- Diagonalizability — conditions, diagonal matrices
- Gershgorin disk theorem
- Commuting operators
- Matrix limits and Markov chains
- Invariant subspaces and the Cayley–Hamilton theorem