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.