Linear Algebra/Jordan Form

This section uses material from three optional subsections: Direct Sum, Determinants Exist, and Other Formulas for the Determinant.

The chapter on linear maps shows that every $$h:V\to W$$ can be represented by a partial-identity matrix with respect to some bases $$B\subset V$$ and $$D\subset W$$. This chapter revisits this issue in the special case that the map is a linear transformation $$t:V\to V$$. Of course, the general result still applies but with the codomain and domain equal we naturally ask about having the two bases also be equal. That is, we want a canonical form to represent transformations as $${\rm Rep}_{B,B}(t)$$.

After a brief review section, we began by noting that a block partial identity form matrix is not always obtainable in this $$B,B$$ case. We therefore considered the natural generalization, diagonal matrices, and showed that if its eigenvalues are distinct then a map or matrix can be diagonalized. But we also gave an example of a matrix that cannot be diagonalized and in the section prior to this one we developed that example. We showed that a linear map is nilpotent&mdash; if we take higher and higher powers of the map or matrix then we eventually get the zero map or matrix&mdash; if and only if there is a basis on which it acts via disjoint strings. That led to a canonical form for nilpotent matrices.

Now, this section concludes the chapter. We will show that the two cases we've studied are exhaustive in that for any linear transformation there is a basis such that the matrix representation $${\rm Rep}_{B,B}(t)$$ is the sum of a diagonal matrix and a nilpotent matrix in its canonical form.