diff --git a/src/cplx-eigenvals.xml b/src/cplx-eigenvals.xml index 4c5d52c..8faa49e 100644 --- a/src/cplx-eigenvals.xml +++ b/src/cplx-eigenvals.xml @@ -29,14 +29,14 @@ the license is included in gfdl.xml.

- In , we saw that an n \times n matrix whose characteristic polynomial has n distinct real roots is diagonalizable: it is similar to a diagonal matrix, which is much simpler to analyze. The other possibility is that a matrix has complex roots, and that is the focus of this section. It turns out that such a matrix is similar (in the 2\times 2 case) to a rotation-scaling matrix, which is also relatively easy to understand. + In , we saw that an n \times n matrix whose characteristic polynomial has n distinct roots is diagonalizable: it is similar to a diagonal matrix, which is much simpler to analyze. The other possibility is that a matrix has complex roots, and that is the focus of this section. It turns out that such a matrix is similar (in the 2\times 2 case) to a rotation-scaling matrix, which is also relatively easy to understand.

In a certain sense, this entire section is analogous to , with rotation-scaling matrices playing the role of diagonal matrices.

- In , we saw that a matrix whose characteristic polynomial has distinct real roots is diagonalizable: it is similar to a diagonal matrix, which is much simpler to analyze. In this section, we study matrices whose characteristic polynomial has complex roots. If a matrix has distinct complex eigenvalues, then it is also diagonalizable, but it similar to a diagonal matrix with complex entries. The geometric interpretation of such a matrix is a subtle question, which is treated in detail in the full version of the book. + In , we saw that a matrix whose characteristic polynomial has distinct roots is diagonalizable: it is similar to a diagonal matrix, which is much simpler to analyze. In this section, we study matrices whose characteristic polynomial has complex roots. If a matrix has distinct complex eigenvalues, then it is also diagonalizable, but it similar to a diagonal matrix with complex entries. The geometric interpretation of such a matrix is a subtle question, which is treated in detail in the full version of the book.