If a matrix is not defective, you can use its eigenvectors as new
basis. It turns out that in that basis the matrix simplifies to a
diagonal matrix
(15.4) |
Needless to say, this simplification is a tremendous help if you are doing analytical or numerical work involving the matrix.
The quickest was to see why is diagonal like above is to note
that in terms of the new basis, produces a new vector
according to
Recall from section 15.1 that the transformation
matrix for change of basis to the eigenvectors must equal the
matrix of eigenvectors. You therefore have for any vector and matrix that you want to transform from new coordinates to
old or vice-versa:
(15.5) |
In summary, a nondefective matrix becomes diagonal when its eigenvectors are used as basis. The main diagonal contains the eigenvalues, ordered like the corresponding eigenvectors in .