Thread: [SOLVED] Linear Algebra, Nilpotent

Let be a finite dimensional vector space over . Let be a linear transformation
on all of whose eigenvalues are zero. Show that for some , i.e.
that is nilpotent.

Originally Posted by JJMC89

Let be a finite dimensional vector space over . Let be a linear transformation
on all of whose eigenvalues are zero. Show that for some , i.e.
that is nilpotent.

Do you know the Cayley-Hamilton theorem?

Originally Posted by JJMC89

Let be a finite dimensional vector space over . Let be a linear transformation
on all of whose eigenvalues are zero. Show that for some , i.e.
that is nilpotent.

http://www.mathhelpforum.com/math-he...nilpotent.html

Originally Posted by dwsmith

http://www.mathhelpforum.com/math-he...nilpotent.html

This is not the same.

In your topic, you wanted to show that if A is nilpotent, then all of its eigenvalues are 0.

Here he wants to prove that if all of A's eigenvalues are 0, then it is nilpotent.

Originally Posted by Defunkt

Do you know the Cayley-Hamilton theorem?

Yes.

Well, then you have it, don't you? If all eigenvalues are 0, then the characteristic equation is and every matrix satisfies its own characteristic equation.