# Finding the Characteristic Polynomial of a Map

• Mar 13th 2011, 11:57 AM
h2osprey
Finding the Characteristic Polynomial of a Map
Let $\displaystyle A: V \rightarrow V$ be a linear map satisfying the property $\displaystyle A^2 = A$. Assume that $\displaystyle dim V = n, rank A = k.$ Find the characteristic polynomial of $\displaystyle A.$

I don't really know where to start, is this an application of the Cayley-Hamilton Theorem or something?

Thanks!
• Mar 13th 2011, 12:37 PM
Opalg
Quote:

Originally Posted by h2osprey
Let $\displaystyle A: V \rightarrow V$ be a linear map satisfying the property $\displaystyle A^2 = A$. Assume that $\displaystyle dim V = n, rank A = k.$ Find the characteristic polynomial of $\displaystyle A.$

I don't really know where to start, is this an application of the Cayley-Hamilton Theorem or something?

Hint: The image of A is the eigenspace for the eigenvalue 1 (and has dimension k). The null space of A is the eigenspace for the eigenvalue 0 (and has dimension n–k).
• Mar 13th 2011, 02:25 PM
h2osprey
Of course, it's all clear now. Thanks!