Week 4 - Diagonalisation, Linear systems of recurrence equations & differential equations

Question 1

2 conditions for a square matrix A to be diagonalisable

Answer 1

1. There is an invertible matrix P
2. & a diagonal matrix D such that
   P^-1AP = D so AP = PD

An nxn matrix is diagonalisable if and only if it has n {enough} LI eigenvectors

Question 2

What does it mean if there are “distinct eigvenvalues”?

Provide the proof for the Theorem: If an nxn matrix has n distinct evalues, then it is diagonalisable.

Answer 2

1. Eigenvectors corresponding to distinct eigenvalues are LI
2. If A has n distinct evalues, then A has n LI evectors, then P will be invertible, then A is diagonalisable.
   *but does NOT work the other way. If no n distinct evalues, doesn’t mean matrix is not diagonalisable, still depends on whether there are enough (n) LI evectors!