## Key Sequence

- we defined an eigenspace, which is the space of all eigenvalues of a distinct eigenvector, and show that
- they form a direct sum to the whole space
- and, as a correlate to how direct sums are kind of like disjoint sets, we have the perhaps expected result of dimension, that the sum of the eigenspace’ dimensions must be smaller than or equal than that of \(V\)

- we defined a Diagonal Matrix, which by its structure + calculation can be shown to require that it is formed by a basis of eigenvalues
- and from there, and the properties of eigenspaces above, we deduce some conditions equal to diagonalizability
- a direct correlary of the last point (perhaps more straightforwardly intuited by just lining eigenvalue up diagonally in a matrix) is that enough eigenvalues implies diagonalizability

## New Definitions

## Results and Their Proofs

- eigenspaces are a direct sum
- dimension of sum of eigenspaces is smaller than or equal to the dimension of the whole space
- conditions equal to diagonalizability
- enough eigenvalues implies diagonalizability

## Questions for Jana

- for diagonalizability, shouldn’t \(n\) be \(m\) on item 3?

## Interesting Factoids

Short!