Discussion Overview
The discussion revolves around the implications of Axler's spectral theorem concerning normal matrices and diagonalizability in both complex and real inner product spaces. Participants explore whether a linear operator can be diagonalizable with non-orthogonal eigenvectors if it is not normal, and the relationship between normality and the properties of eigenvalues.
Discussion Character
- Debate/contested
- Mathematical reasoning
Main Points Raised
- One participant questions whether a linear operator T can be diagonalizable with non-orthogonal eigenvectors if T is not normal, particularly in complex inner product spaces.
- Another participant suggests constructing a 2x2 example of a diagonalizable matrix with non-orthogonal eigenvectors to explore the conjecture.
- A participant inquires if the spectral theorem guarantees that algebraic and geometric multiplicities for all eigenvalues are equal for normal operators.
- A later reply asserts that in a complex inner product space, every normal operator has a full orthonormal set of basis vectors, while in a real inner product space, every self-adjoint operator does as well.
Areas of Agreement / Disagreement
Participants express differing views on the implications of the spectral theorem, particularly regarding diagonalizability and the properties of eigenvectors in relation to normality. The discussion remains unresolved on some points, particularly the conjecture about diagonalizability without normality.
Contextual Notes
The discussion includes assumptions about the definitions of normal and self-adjoint operators, as well as the conditions under which eigenvalues exist in real inner product spaces. There are unresolved questions about the implications of the spectral theorem in different contexts.