4.3 Diagonalization, Similarity, and Powers of a Matrix
4.3 Diagonalization, Similarity, and Powers of a Matrix
Reading
Work on the preview activity and read Chapter 4 Section 3 in Understanding Linear Algebra by David Austin.
The idea behind diagonalization:
A is diagonalizable if we can write [latex]A= PDP^{-1}[/latex], where D is a diagonal matrix. The columns of P consist of eigenvectors of A and the diagonal entries of D are the associated eigenvalues.
This video has very nice animations/explanations of the change of basis formula behind diagonalization:
Computing diagonalization by hand:
Using Sage to find the diagonalization:
Powers of a matrix via diagonalization:
- A is diagonalizable if and only if it has a basis of eigenvectors.
- If [latex]A=PDP^{-1},[/latex] then [latex]A^k = PD^kP^{-1}[/latex]