An n x n matrix A is called nilpotent if Am = 0 for some positive integer m. Examples
Chapter 3, Problem 76(choose chapter or problem)
An n x n matrix A is called nilpotent if Am = 0 for some positive integer m. Examples are triangular matrices whose entries on the diagonal are all 0. Consider a nilpotent n x n matrix A, and choose the smallest number m such that A = 0. Pick a vector v in Rn such that Am~]v j=. 0. Show that the vec- Am~xv tors are linearly independent. v, Av, A2v, (Hint: Consider a relation cov + c\Av + C2A2v -\------h cm-\A m- xv = 0. Multiply both sides of the equation with Am~x to show that co = 0. Next, show that ci = 0, and so on.)
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