example of non-diagonalizable matrices
Some matrices with real entries which are not diagonalizable over ℝ are diagonalizable over the complex numbers ℂ.
For instance,
A=(0-110) |
has λ2+1 as characteristic polynomial.
This polynomial
doesn’t factor over the reals, but over ℂ it does. Its roots are λ=±i.
Interpreting the matrix as a linear transformation ℂ2→ℂ2, it has eigenvalues i and -i and linearly independent
eigenvectors
(1,-i), (-i,1). So we can diagonalize A:
A=(0-110)=(1-i-i1)(i00-i)(.5.5i.5i.5) |
But there exist real matrices which aren’t diagonalizable even if complex eigenvectors and eigenvalues are allowed.
For example,
B=(0100) |
cannot be written as UDU-1 with D diagonal.
In fact, the characteristic polynomial is λ2 and it has only one double root λ=0.
However the eigenspace corresponding to the 0 (kernel) eigenvalue has dimension
1.
B(v1v2)=(00)⇔v2=0 and thus the eigenspace is ker(B)=spanℂ{(1,0)T}, with only one dimension.
There isn’t a change of basis where B is diagonal.
Title | example of non-diagonalizable matrices |
---|---|
Canonical name | ExampleOfNondiagonalizableMatrices |
Date of creation | 2013-03-22 14:14:30 |
Last modified on | 2013-03-22 14:14:30 |
Owner | cvalente (11260) |
Last modified by | cvalente (11260) |
Numerical id | 14 |
Author | cvalente (11260) |
Entry type | Example |
Classification | msc 15-00 |