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I've some questions from the following theorem:

Let $T:V\to V$ be a nilpotent linear operator with index of nilpotency $k.$ Then $T$ can be expressed as a block diagonal matrix representation where each block $N$ is a Jordan nilpotent matrix. Moreover there is at least one such $N$ of order $k$ and all other $N$ are of order $\le k.$ Moreover the no of $N$ of order $i(\le k)$ is uniquely determined by $T$ and total no of $N$ of all order is $\ker T.$

$$Now~my~questions~are: $$

  1. Does such representation of $T$ look like this? enter image description here

  2. What is meant by "the no of $N$ of order $i(\le k)$ is uniquely determined by $T$"? Does it mean if $T$ is represented by two different such (Jordan nilpotent) block diagonal matrices then the no of block matrices of order $i$ remain same in those two?

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1 Answer 1

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Yes, the matrix looks like that. Note that at least one of the blocks is notjust $\le k\times\le k$ but in fact $k\times k$.

And Yes, two different such block matrices do have the "same" blocks, but possibly permuted.

The claims can be easily verified by looking at the numbers $\dim \ker T^i$ for $i=1, \ldots, k$.

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