- Review the box
- Algebraic multiplicity bounds geometric multiplicity. Take a basis for the eigenspace. Extend.
- Let
*T*be a linear operator and*λ*_{i}be distinct eigenvalues. For each*λ*_{i}, let*v*_{i}∈*E*_{λi}, then if ∑*v*_{i}= 0, each*v*_{i}is zero. - Unions of linearly independent subsets from eigenspaces are still linearly independent.
- Diagonalizable if and only if algebraic multiplicity equals geometric multiplicity.
Proof: Let

*m*_{i}be algebraic multiplicity,*d*_{i}be geometric multiplicity, and*n*= dim*V*.Suppose diagonalizable. Then let B be a basis consisting of eigenvectors. Intersect this basis with each eigenspace. Call it

*B*_{i}. Let*n*_{i}=*B*_{i}. Then*n*≤ ∑*n*_{i}≤ ∑*d*_{i}≤ ∑*m*_{i}≤*n*.So ∑(

*m*_{i}−*d*_{i}) = 0. But each term must be nonnegative.Conversely, suppose the multiplicties are equal. Then take union of eigenbasis. They form a basis for full space.

- Testing diagonalizability. Check splitting, check multiplicities. This also computes the diagonalization along the way.
- What are eigenvalues of derivative?
- Finite dimensional matrices always have eigenvalues if you extend the field.
- Give example of infinite dimension transform with no eigenvalues.
- Work through example 6 in book.

- Define direct sum
- Give example
- Show Jasper’s example to motivate
- Diagonalizable if and only if direct sum of eigenspaces
- The following are equivalent:
- direct sum
- sums and unique way of writing 0
- unique way of writing any element
- union of ordered basis for summands is ordered basis
- exists order basis fof summands so that union is ordered basis.

- Do more examples