------5.1
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.
5.2 Diagonalizbility
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.