example of non-diagonalizable matrices

Some matrices with real entries which are not diagonalizable over are diagonalizable over the complex numbersMathworldPlanetmathPlanetmath .

For instance,


has λ2+1 as characteristic polynomialMathworldPlanetmathPlanetmath. This polynomialPlanetmathPlanetmath doesn’t factor over the reals, but over it does. Its roots are λ=±i.

Interpreting the matrix as a linear transformation 22, it has eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath i and -i and linearly independentMathworldPlanetmath eigenvectorsMathworldPlanetmathPlanetmathPlanetmath (1,-i), (-i,1). So we can diagonalize A:


But there exist real matrices which aren’t diagonalizable even if complex eigenvectors and eigenvalues are allowed.

For example,


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 eigenspaceMathworldPlanetmath corresponding to the 0 (kernel) eigenvalue has dimensionPlanetmathPlanetmath 1.

B(v1v2)=(00)v2=0 and thus the eigenspace is ker(B)=span{(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