Discussion Overview
The discussion revolves around the identification and computation of diagonalizable 3x3 matrices over the finite field GF(2), particularly in the context of graph polynomial research. Participants explore the properties of these matrices, including their eigenvalues and the implications of diagonalizability.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
- Mathematical reasoning
Main Points Raised
- One participant seeks to compute the 8 diagonalizable 3x3 matrices over GF(2) without checking all 512 possibilities.
- Another participant suggests that diagonal matrices have only non-zero entries on the diagonal, leading to 8 possible diagonal matrices, but notes that eigenvalues cannot be distinct due to the limited values in GF(2).
- Some participants argue about the validity of the spectral theorem's application to diagonalizability, with one stating that diagonalizable matrices need not be symmetric.
- There is a discussion about the relationship between diagonalizability and the existence of orthogonal eigenbases, with some participants clarifying that diagonalizable matrices do not require orthogonal eigenvectors.
- One participant mentions that there are more than 8 diagonalizable matrices, asserting that all diagonal matrices in GF(2) are projections.
- Another participant highlights a disconnect regarding whether the original inquiry was about diagonalizable matrices or diagonalizable invertible matrices.
- One participant shares results from a program counting invertible and diagonalizable matrices, noting surprising numbers (168 invertible and 58 diagonalizable matrices) and providing a counterexample of a symmetric matrix that is not diagonalizable.
- Confusion arises regarding the definitions of diagonal and diagonalizable matrices, with clarifications provided by participants.
Areas of Agreement / Disagreement
Participants express differing views on the properties of diagonalizable matrices, particularly regarding the spectral theorem, the necessity of distinct eigenvalues, and the definitions of diagonal versus diagonalizable. The discussion remains unresolved with multiple competing views on these topics.
Contextual Notes
Participants note limitations in their understanding of the definitions and properties of matrices over GF(2), as well as the implications of diagonalizability in this context. There are also unresolved questions about the computational methods for identifying diagonalizable matrices.