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.
Exercise 1.
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
(1)
( is an "eigenvector" up to a root vector of lower order). Similarly, denoting
we have
(2)
...
Continuing in the same way, we get
(3)
(4)
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
(5)
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.