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Section 7.5 Cyclic subspaces

Definition 7.5.1. (Principal vector and its order).

Let \(V\) be a finite-dimensional vector space over a field \(F\text{.}\) A nonzero vector \(e\in V\) is said to be a principal vector of an \(F\)-linear map \(T\colon V\to V\) with an eigenvalue \(\lambda\in F\) if there exists a natural number \(n\) such that \((T-\lambda\unit_V)^n(e)=0\text{.}\)

The least natural number \(m\) such that \((T-\lambda\unit_V)^m(e)=0\) is called the order of \(e\text{.}\)

A principal vector of \(T\) with an eigenvalue \(\lambda\) is also called a generalized eigenvector with eigenvalue \(\lambda\).

Remark 7.5.2.
An eigenvector is a principal vector of order \(1\text{.}\)
Let \(T\colon\R^2\to\R^2\) be a linear map corresponding to a matrix \(A=\begin{pmatrix}1\amp 1\\0\amp 1\end{pmatrix}\) (see Theorem 5.2.3). We have
\begin{equation*} (T-\unit_V)(e_1)=0\quad\text{and}\quad(T-\unit_V)(e_2)=e_1. \end{equation*}
Thus,
\begin{equation*} (T-\unit_V)^2(e_2)=0. \end{equation*}
Therefore, \(e_1\) is a principal vector of \(T\) with eigenvalue \(1\) and its order is \(1\text{.}\) While \(e_2\in\R^2\) is a principal vector of \(T\) with an eigenvalue \(1\text{,}\) and its order is \(2\text{.}\) Since vectors \(w_1=e_2,w_2=(T-\unit_V)(e_2)\) are \(\R\)-linearly independent, they form a basis of \(\R^2\text{.}\) Furthermore, since \(T(w_1)=w_1+w_2\) and \((T-\unit_V)(w_2)=(T-\unit_V)^2(e_2)=0\text{,}\) the matrix of \(T\) with respect to an ordered basis \((w_1,w_2)\) is
\begin{equation*} \begin{pmatrix}1\amp0\\1\amp1\end{pmatrix}. \end{equation*}
Definition 7.5.4. (Cyclic subspace).
Let \(V\) be a finite-dimensional vector space over a field \(F\text{.}\) Consider a nonzero (principal) vector \(e\in V\text{,}\) a scalar \(\lambda\in F\) such that \((T-\lambda\unit_V)^n(e)=0\text{.}\) Suppose that \(m\) is the least natural number such that \((T-\lambda\unit_V)^m(e)=0\text{.}\) The subspace of \(V\) generated by
\begin{equation*} e,(T-\lambda\unit_V)(e),(T-\lambda\unit_V)^2(e),\ldots,(T-\lambda\unit_V)^{m-1}(e) \end{equation*}
is called the cyclic subspace generated by \(e\).

We first compute the dimension of \(W\text{.}\) By Definition 7.5.4, the subspace \(W\) is spanned by

\begin{equation*} e,\;(T-\lambda\unit_V)(e),\;(T-\lambda\unit_V)^2(e),\ldots,(T-\lambda\unit_V)^{m-1}(e). \end{equation*}

Hence, \(\dim_FW\leq m\text{.}\) We show that \(\left\{(T-\lambda\unit_V)^i(e)\right\}_{i=0}^{m-1}\) is linearly independent. Suppose that

\begin{equation*} a_0e+a_1(T-\lambda\unit_V)(e)+\cdots+a_{m-1}(T-\lambda\unit_V)^{m-1}(e)=0. \end{equation*}

Since \((T-\lambda\unit_V)^m(e)=0\text{,}\) we get that \((T-\lambda\unit_V)^r(e)=0\) for any \(r\geq m\text{.}\) Hence, the linear map \((T-\lambda\unit_V)^{m-1}\) evaluated at the above expression gives

\begin{equation*} a_0(T-\lambda\unit_V)^{m-1}(e)=0. \end{equation*}

As \((T-\lambda\unit_V)^{m-1}(e)\neq 0\text{,}\) we have \(a_0=0\text{.}\) Therefore we are left with

\begin{equation*} a_1(T-\lambda\unit_V)(e)+\cdots+a_{m-1}(T-\lambda\unit_V)^{m-1}(e)=0. \end{equation*}

Now we apply \((T-\lambda\unit_V)^{m-2}\) to the above expression, and argueing as above, we obtain \(a_1=0\text{.}\) Continuing in this way we get that \(a_i=0\) for all \(i\text{.}\) This shows that \(\left\{(T-\lambda\unit_V)^i\right\}_{i=0}^{m-1}\) is a maximal linearly independent subset of \(W\text{,}\) i.e., it is a basis of \(W\text{.}\) Thus \(\dim_FW=m\text{.}\)

We now show that \(W\) is invariant under \(T\text{.}\) We rename the basis vectors of \(W\) obtained above as follows.

\begin{equation*} w_1=e,\;w_2=(T-\lambda\unit_V)(e),\;w_3=(T-\lambda\unit_V)^2(e),\;\ldots,\;w_m=(T-\lambda\unit_V)^{m-1}(e) \end{equation*}

Therefore we have

\begin{equation*} (T-\lambda\unit_V)(w_i)=w_{i+1}\quad\text{for}\;1\leq i\leq m-1\quad\text{and}\quad (T-\lambda\unit_V)(w_m)=0. \end{equation*}

In other words we have

\begin{equation*} T(w_i)=\lambda w_i+w_{i+1}\in W\quad\text{for}\;1\leq i\leq m-1\quad\text{and}\quad T(w_m)=\lambda w_m\in W. \end{equation*}

This shows that \(W\) is invariant under \(T\text{.}\)

Remark 7.5.6. (Matrix of a cyclic subspace).
We keep the notation of the above Proposition 7.5.5 and its proof. A matrix of \(T|_W\) with respect to the ordered basis \((w_1,w_2,\ldots,w_m)\) is the matrix with all diagonal entries \(\lambda\text{,}\) all lower diagonal entries \(1\text{,}\) and all other entries \(0\text{.}\)
Definition 7.5.7. (Jordan block).
The matrix of a cyclic subspace described in Remark 7.5.6 is called the Jordan block. Thus Jordan block obtained in Proposition 7.5.5 is the following lower triangular matrix.
\begin{equation*} J_\lambda=\begin{pmatrix}\lambda\amp 0\amp 0\amp\cdots\amp\cdots\amp 0\\1\amp\lambda\amp 0\amp\amp\amp 0\\0\amp 1\amp\lambda\amp\amp\amp\vdots\\\vdots\amp\amp\ddots\amp\ddots\amp\amp\vdots\\\vdots\amp\amp\amp\ddots\amp\lambda\amp 0\\0\amp 0\amp\cdots\amp\cdots\amp 1\amp \lambda\end{pmatrix} \end{equation*}
Remark 7.5.8.
The Jordan block of size \(1,2,3\) and \(4\) are, respectively
\begin{equation*} (\lambda),\;\begin{pmatrix}\lambda\amp\\1\amp\lambda\end{pmatrix},\;\begin{pmatrix}\lambda\amp \amp \\1\amp\lambda\amp \\\amp 1\amp\lambda\end{pmatrix},\;\begin{pmatrix}\lambda\amp\amp\amp\\1\amp\lambda\amp\amp\\\amp 1\amp\lambda\amp\\\amp\amp1\amp\lambda\end{pmatrix}. \end{equation*}
This follows from the properties of determinant. The third statement is consequence of Lemma A.1.2. We now show the last statement. Suppose that \(J_\lambda\in M_r(F)\text{.}\) Note that \(J_\lambda=\lambda I_r+J_0\text{.}\) We have \(J_0^k=0\) for all \(k\geq r\text{.}\) Since \(\lambda I_r\cdot J_0=J_0\cdot\lambda I_r\) we have
\begin{equation*} \left(J_r-\lambda I_r\right)^k=J_0^k. \end{equation*}
By Proposition 7.2.20, the minimal polynomial divides the characteristic polynomial hence, we get the result.