The Jordan-Chevalley Decomposition
We recall that any linear endomorphism of a finite-dimensional vector space over an algebraically closed field can be put into Jordan normal form: we can find a basis such that its matrix is the sum of blocks that look like
where is some eigenvalue of the transformation. We want a slightly more abstract version of this, and it hinges on the idea that matrices in Jordan normal form have an obvious diagonal part, and a bunch of entries just above the diagonal. This off-diagonal part is all in the upper-triangle, so it is nilpotent; the diagonalizable part we call “semisimple”. And what makes this particular decomposition special is that the two parts commute. Indeed, the block-diagonal form means we can carry out the multiplication block-by-block, and in each block one factor is a constant multiple of the identity, which clearly commutes with everything.
More generally, we will have the Jordan-Chevalley decomposition of an endomorphism: any can be written uniquely as the sum , where is semisimple — diagonalizable — and is nilpotent, and where and commute with each other.
Further, we will find that there are polynomials and — each of which with no constant term — such that and . And thus we will find that any endomorphism that commutes with with also commute with both and .
Finally, if is any pair of subspaces such that then the same is true of both and .
We will prove these next time, but let’s see that this is actually true of the Jordan normal form. The first part we’ve covered.
For the second, set aside the assertion about and ; any endomorphism commuting with either multiplies each block by a constant or shuffles similar blocks, and both of these operations commute with both and .
For the last part, we may as well assume that , since otherwise we can just restrict to . If then the Jordan normal form shows us that any complementary subspace to must be spanned by blocks with eigenvalue . In particular, it can only touch the last row of any such block. But none of these rows are in the range of either the diagonal or off-diagonal portions of the matrix.