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— if we take higher and higher powers of the map or matrix then we eventually get the zero map or matrix— 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.