# Strictly Upper Triangular Matrix Proof Help Needed

• May 22nd 2008, 08:56 PM
Ultros88
Strictly Upper Triangular Matrix Proof Help Needed
Hey, I'm trying to prove that for a strictly upper triangular matrix A, A^n = 0. I'm pretty much stuck at writing down the definition for matrix multiplication for A*A. Any help/hints would be much appreciated. Thanks, Ultros
• May 22nd 2008, 09:51 PM
mr fantastic
Quote:

Originally Posted by Ultros88
Hey, I'm trying to prove that for a strictly upper triangular matrix A, A^n = 0. I'm pretty much stuck at writing down the definition for matrix multiplication for A*A. Any help/hints would be much appreciated. Thanks, Ultros

A is nilpotent. Show either that its eigenvalues are all zero or that its determinant is zero.
• May 23rd 2008, 11:35 AM
Ultros88
Proof Excercise Comes Before Nilpotents/Determinants
Is there a way to prove that A^n = 0 without using the fact that its determinant or eigenvalues are all zero? This exercise comes before those topics in the book that I am using. Thanks
• May 23rd 2008, 02:32 PM
mr fantastic
Quote:

Originally Posted by Ultros88
Is there a way to prove that A^n = 0 without using the fact that its determinant or eigenvalues are all zero? This exercise comes before those topics in the book that I am using. Thanks

Proof by induction will work (you have to know how to do matrix multiplication). See here: Matrix Algebra - Google Book Search