Eigenvalues + Algebraic/Geometric Multiplicity

[verd]

I'm studying for a linear algebra final, and I'm looking over an old final our prof gave us and I've come across something I don't remember ever hearing anything about... Here's the problem:

Write down a matrix A for the following condition:
A is a 3x3 matrix with lambda=4 with algebraic multiplicity 3 and with geometric multiplicity 1.

...I don't have a problem with eigenvalues or anything, but I don't believe he ever mentioned algebraic multiplicity or geometric multiplicity. Is this another concept in linear algebra?? Or is this something way simple that I'm looking way too far into.

...What does he mean by algebraic multiplicity and geometric multiplicity??


Thanks!

 

[Dick]

Algebraic multiplicity is easy. It's the multiplicity of the root in the characteristic polynomial. I checked wikipedia (always a good first stab) for geometric multiplicity and it says that it is the dimension of the eigenspace. In other words, there is only one linearly independent eigenvector with value 4.

 

[boof]