Jordan Canonical Forms, An Introduction


Jordan (biography) Looked at linear transformations from a vector space into itself, and wondered how many "different" transformations were possible. The key here is the word different. Two transformations are considered the same if one looks just like the other after an appropriate change of basis. Scaling x by 2 is really no different than scaling y by 2; you just have to interchange the axes. In contrast, these transformations are nothing like the map that squashes the plane down to the origin. Choose any basis you like; scaling x by 2 just doesn't look like compressing the plane to a point.

As you recall, two matrices are similar if one is equal to the other after an appropriate change of basis. Similar matrices fall into equivalence classes, and Jordan wanted to understand these classes, and select a canonical representative for each.

As we shall see, any matrix M is similar to a matrix J, where J is a diagonal matrix with some scattered ones along the subdiagonal (just below the main diagonal). Here J is the canonical representative for M, or more precisely, the equivalence class of matrices similar to M. Choose a different coordinate system, and the transformation that was implemented by M is now implemented by J, a much simpler matrix. This works in any field, although you may need to extend the field to its algebraic closure. The new basis, acted upon by J, and the diagonal elements of J, may employ elements from the algebraic closure.

This theorem is almost a corollary when M is diagonalizable. Let J be the diagonal matrix that is similar to M, and J is in jordan canonical form.

When M is not diagonalizable we have more work to do. Let's get started.