- eigenvectors are invariant under change of basis.
- charpoly is invariant under change of basis.
- A matrix is diagonalizable if and only if there is a basis for Fn consisting of eigenvectors for A.
- Conditions for when a matrix is diagonalizable
- Most of the time is it over CC.
- always diagonalizable with distinct eigenvalues.
- Split over F means you have n roots over F
- A polynomial with coefficients over C splits over C.
- If diagonalizable, then the charpoly splits.
- The algebraic multiplicity is a thing
- Eigenspace and geometric multiplicity is a thing
- Algebraic mulitplicty bounds geometric multiplicity
- sum of Eigenspaces give direct sum
- unions of linearly independent subsets of eigenspaces are still linearly independent.
- Diagonalizble if and only if algebraic mulitplicity equals geometric multiplicity.