Constructing invariant subspaces from scratch. An algorithm So, basically i am trying to prove that there exist a basis w.r.t which there exists an upper triangular matrix in a complex field.
Most of the books which i read incorporate induction as a method which i find sadly non intuitive. Please have a look at the following :
Suppose in the complex field, I select an eigen vector $v_1$ such that $Tv_1 = kv_1$.
Now, i must select a vector $v_2$ such that:
(a) $v_1$ and $v_2$ are linearly independent.
(b) $Tv_2$ lies in span$(v_1,v_2)$.
Writing mathematically, $v_2$ should be such that
(i) $rv_1 + sv_2 = 0 \implies r = s = 0$.
(ii) $Tv_2 = a v_1 + b v_2 \implies (T - b I )v_2 = av_1$
What can I infer for $v_2$ from here?
Some points which may be helpful like:
$av_1$ lies in range $(T - b I)$
Now since the range of any linear mapping is a subspace,
$v_1$ also lies in the range $(T - b I)$.
Help will be really appreciated as I have spent a long time analyzing this. Thank you.
 A: Well, think about the linear transformation an upper triangular matrix 
$$\begin{pmatrix}
\lambda_1 & \ast & \ast \\
0 & \lambda_2 & \ast \\
0 & 0 & \lambda_3 \end{pmatrix}$$
represents.
It takes $e_3$ to $\lambda_3 e_3$ -- that is, it fixes $\operatorname{span} \{e_3\}$.  It takes $e_2$ to a linear combination of $e_2$ and $e_3$; since it also takes $e_3$ to a linear combination of $e_2$ and $e_3$ (specifically, a linear combination where the coefficient of $e_2$ is zero!), this means it also fixes $\operatorname{span} \{e_2, e_3\}$.  And of course it fixes $\operatorname{span} \{e_1, e_2, e_3\}$, which in this case is not saying anything because that's the entire space.
So, a matrix in upper triangular form gives us


*

*a line fixed by the transformation; 

*a plane fixed by the transformation and containing that line;

*a three-space fixed by the transformation and containing that plane;

*and so forth up to the dimension of $n$.


Consequently, to put a matrix in upper triangular form, we should reverse this process by finding such a line, then such a plane, etc.
