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I'm trying to visualize what a linear operator does to a vector space if that operator can be put into Jordan normal form.

For concrete motivation, let's take $V = \mathbb{R}^3$, with some linear operator $T: V \rightarrow V$, and let's assume $T$ can be written as an upper-triangular matrix $M$. Then $T$ stabilizes a complete flag: there's a nested sequence of invariant subspaces $U \subset W \subset V$, with $\dim U = 1$ and $\dim W = 2$. Geometrically, $T$ fixes a line and a plane containing that line. This already tosses out lots of transformations; I would like to know what possibilities are left.

For starters, I think $T$ can't be a rotation around a line contained in $W$, since that would move $W$, and it also can't be a rotation around a line passing through $W$, since that would move $U$; so apparently $T$ can't be a rotation at all (except possibly by $\pi$ or $2\pi$). However, it seems like certain combinations of shearing, reflecting and scaling would work, although even there I must be careful. (The exact combinations allowable seem complicated.) Is this it? Am I missing anything?

Suppose I further want $M$ to be diagonalizable. If I 'pin down' another line in $W$, I think this would rule out shearing of $W$ along any line it contains (although shearing of $V$ along the plane $W$ would still be OK) and $M$ would become slightly more diagonal: a 2x2 upper-triangular matrix and an eigenvalue sitting in the corner. Finally, if I 'pin down' a line passing through $W$, I think it would toss out shearing of $V$ altogether and $M$ would finally be diagonal. Is this right?

Next, let's turn to a general n-dimensional $V$ with $T: V \rightarrow V$, and assume its $n$ x $n$ matrix $M$ is Jordan. Then $T$ stabilizes a flag; but also, every $m$ x $m$ Jordan block corresponds to some $m \le n$ - dimensional invariant subspace which itself contains a nested sequence of invariant subspaces, so I think $T$ in fact stabilizes many different flags! Is this right? Is it correct to say that $T$ would then act on each of these subspaces as described above? Can we describe the process of diagonalizing $T$, if possible, as equivalent to finding more and more invariant subspaces of progressively smaller dimension (eventually dim 1), collapsing each Jordan block as we go and ruling out more and more transformations (rotations, shears, etc.) until all that's left is scaling along each 1-dimensional subspace?

If this is indeed a correct visualization, does it generalize at all to infinite-dimensional vector spaces?

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I personally like to imagine what happens in a single block when the eigenvalue $\lambda = 0$. It keeps projecting down to lower and lower subspaces in a chain. In the case $\lambda \neq 0$, we're adding $\lambda I$ to each step so we keep shearing it in a way. But yes, it stabilises each flag corresponding to a jordan block. – muzzlator Mar 19 '13 at 12:22

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