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
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