Action of a matrix in its root subspace
The purpose of the following discussion is to reveal the matrix form of in
Definition 1. Nonzero elements of are called root vectors. This definition can be detailed as follows:
Elements of are eigenvalues.
Elements of are called root vectors of 1st order.
Elements of are called root vectors of order .
Thus, root vectors belong to
where the sets of root vectors of different orders do not intersect.
Proof. Suppose that is, and Denoting we have and which means that and maps into
Now, starting from some we extend a chain of root vectors all the way to an eigenvector. By Exercise 1, the vector belongs to From the definition of we see that
( is an "eigenvector" up to a root vector of lower order). Similarly, denoting we have
Continuing in the same way, we get
Exercise 2. The vectors defined above are linearly independent.
Proof. If then Here the left side belongs to and the right side belongs to because of inclusion relations. Hence, Similarly, all other coefficients are zero.
By Exercise 2, the vectors form a basis in
Exercise 3. The transformation in is given by the matrix
where is a matrix with unities in the first superdiagonal and zeros everywhere else.
Proof. Since is taken as the basis, can be identified with the unit column-vector The equations (1)-(4) take the form
Putting these equations side by side we get
Definition 2. The matrix in (5) is called a Jordan cell.