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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 A=PDP−1, 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 A=PDP−1, then Ak=PDkP−1

 

License

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Math 220, Matrices Copyright © 2018 by Kristen Pueschel is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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