## 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**.