Diagonalisable
Definition
-if for a square matrix A there exists an invertible matrix P such that P^-1AP is diagonal then we say that A is diagonalisable
How to find P for a square matrix A such that
P^-1AP = D
- find the eigenvalues and eigenvectors of the matrix A
- D is a square matrix with all zero entries except the main diagonal whose values are the eigenvalues of A i.e. d(11) = λ1, d(22) = λ2, d(33) = λ3 etc.
- P is a matrix whose columns are composed of the eigenvectors of A, P = (v1 v2 v3 …)
How can you tell that a matrix is not diagonalisable?
-if there are not enough linearly independent eigenvectors to form a square matrix, P, then we cannot diagonalise the matrix A
Eigenspace
-let A be an nxn square matrix with eigenvalue λ
-we call
Uλ = {v| Av=λv, v∈ℝ^n}
the eigenspace of λ
i.e. Uλ is the set of eigenvectors with eigenvalue λ for A together with the 0 vector
Eigenspaces and Subspaces
-the eigenspace Uλ is a subspace of ℝ^n
Eigenvectors of A and Linearly Independence
- suppose A is an nxn square matrix with at least m distinct eigenvalues with corresponding eigenvectors
- then the set {v1, v2, …, vm} is linearly independent
- proof by induction
Diagonalisable Matrices, Calculating A^m
P^-1*A*P = D A = P*D*P^-1 A^m = (P*D*P^-1) A^m = P*D^m*P^-1