By Eckhard M.S. Hitzer, University of Fukui, Prof. Ryosuke Nagaoka (University of the Air, Japan)

Yes, please have a look at this pattern here. Let us assume that we have the matrix A to be nilpotent, so the mth power of A is supposed to be zero, but smaller powers, (like) A to the power of m minus one is supposed not to be zero. Then we take the characteristic polynomial of the matrix A here, and in this polynomial we replace x by the matrix A itself and get a matrix polynomial. This matrix polynomial is the zero matrix according to the Cayley-Hamilton theorem. Now we take this expression and multiply it by A to the power of m minus one from the right. And the result is A to the power of n plus m minus one, plus a one A to the power of n plus m minus two, plus a n minus one A to the power of m, plus a n A to the power of m minus one equals zero. Because of the nilpotent property of A we have this to be the zero matrix and all higher powers of A also to be the zero matrix. We are only left with this expression here and according to our assumption A to the power of m minus one is not zero. Therefore the scalar coefficient a n must be zero here.

Next we have now an expression, which is shorter. And I write it once more. So A to the power of n, plus a one A to the power of n minus one, plus ... a n minus one A plus zero is zero. Now we multiply once more from the right with A to the power of m minus two. And we get A to the power of n plus m minus two, plus a one A to the power of n plus m minus three, plus ... a n minus one A to the power of m minus one, and before we have the expression a n minus two A to the power of m to be zero. Now according to the nilpotent property, A to the power of m is the zero matrix. And all higher powers of A are zero matices. So we are left with this expression. And according to the assumption, A to the power of m minus one should not be zero. Therefore the coefficient a n minus one must also be a scalar zero.

Now we have proved that the coefficient a n equals zero, the coefficient a n minus one equals zero, and we can continue the same strategy of proof until we find that also the coefficient a one is zero. After learning that all the coefficients must be zero, we are only left with this one expression, A to the power of n must also be zero, the zero matrix. And therefore we have a proof for the theorem, which we were looking for.

Soli Deo Gloria. Created by
Eckhard Hitzer (Fukui).

Last Modified 16 January 2004

EMS Hitzer and Prof. R. Nagaoka are not responsible for the content of external internet sites.