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How would you show any matrix in $F^{n\times n}$ is similar to a matrix of the form

$$\begin{pmatrix} R&0\\ 0&S \end{pmatrix}$$

where $R$ is nilpotent and $S$ is invertible?

Would you designate specific vector spaces and show linear transformations restricted to kernels?

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    $\begingroup$ Is $F$ algebraically closed? If so, try triangularizing your matrix. $\endgroup$ – Alex Youcis Sep 30 '13 at 6:26
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It is a corollary of Fitting lemma.

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Do Jordan decomposition and put the block corresponding to the zero eigenvalue in the left corner.

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  • $\begingroup$ It is not given that $F$ is algebraically closed, so Jordan decomposition might not exist. $\endgroup$ – Marc van Leeuwen Dec 11 '14 at 14:36
  • $\begingroup$ @MarcvanLeeuwen, can i split the space ${0} \subset ker(A) \subset \ker(A^2) \cdots$ write the space $F^n$ as the direct sum of the union of kernels and its complement. now find a basis in each, you get a block representation as needed. $\endgroup$ – abel Dec 11 '14 at 15:10
  • $\begingroup$ Yes, see my answer [Another possible approach] for details. But is won't be a Jordan decomposition. $\endgroup$ – Marc van Leeuwen Dec 11 '14 at 15:12
  • $\begingroup$ @MarcvanLeeuwen thanks for taking time to comment on my answers. the second block is going to be (full) without any structure? $\endgroup$ – abel Dec 11 '14 at 15:28
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Note that the problem is equivalent to finding a decomposition of the vector space $F^n$ into a direct sum $V\oplus W$ of $\phi$-stable subspaces ($\phi$ being the linear operator defined by your matrix) such that the restrictions of $\phi$ to $V$ and to $W$ are nilpotent respectively invertible. Once one has such a decomposition, choosing bases in $V$ and $W$ and expressing $\phi$ on the combined basis suffices.

Now let $P\in K[X]$ be any monic polynomial such that $P[\phi]=0$; you may take it to be the minimal polynomial of$~\phi$, or its characteristic polynomial if you prefer. Write $P=X^mQ$ with $m\geq0$ and $Q$ is not divisible by$~X$. Then since $X^m$ and $Q$ are relatively prime, $F^n=\ker(\phi^m)\oplus\ker(Q[\phi])$, by what many people call the Chinese remainder theorem*. The restriction of $\phi$ to $V=\ker(\phi^m)$ is obviously nilpotent and the restriction of $\phi$ to $W=\ker(Q[\phi])$ is invertible, as $W\cap\ker(\phi)\subseteq W\cap\ker(\phi^k)=W\cap V=\{0\}$.


Alternative ways to see that $\phi|_W$ is invertible are that it cannot have $0$ as eigenvalue, since this is no root of the polynomial$~Q$ annihilating $\phi|_W$, or by explictely computing the inverse as $S[\phi]$ where the polynomial$~S$ is the quotient $(1-c^{-1}Q)/X$ with $c=Q[0]$ the nonzero constant term of$~Q$ (use $XS\equiv1\pmod Q$).

Another possible approach is to consider the kernels of successive powers $\phi^k$. Since the dimensions of the kernels increase weakly with$~k$, it must happen that $\ker(\phi^{k+1})=\ker(\phi^k)$ for some $k\in\Bbb N$. At this point $\def\im{\operatorname{im}}\ker(\phi^k)\cap\im(\phi^k)\subseteq\ker(\phi^k)\cap\im(\phi)=\{0\}$, so the sum $\ker(\phi^k)+\im(\phi^k)$ is direct, and fills the whole space by rank-nullity. Then restriction of $\phi$ to $V=\ker(\phi^k)$ is obviously nilpotent (of order$~k$), and its restriction to $W=\im(\phi^k)$ is surjective since $\im(\phi^{k+1})=\im(\phi^k)$; we have our direct sum decomposition.

*A more appropriate name would be kernel decomposition theorem. The Chinese remainder theorem for (the PID) $F[X]$ just states that any system of congruences $Q\equiv R_i\pmod{P_i}$ ($i=1,\ldots,k$) with pairwise relatively prime polynomials $P_i$ has a unique solution$~Q$ modulo the product$~P=P_1P_2\ldots P_k$. The result needed here is that in this case for any endomorphism$~\phi$ one has $$ \ker(P[\phi]) = \ker(P_1[\phi])\oplus\cdots\oplus\ker(P_k[\phi]) $$ From the Chinese remainder theorem one gets the existence, for $1\leq j\leq n$, of $Q_i\in F[X]$ satisfying $Q_j\equiv \delta_{i,j}\pmod{P_i}$ (for $i=1,\ldots,k$). With $\pi_j$ the restriction to $\ker(P[\phi])$ of $Q_j[\phi]$, it acts by the scalar $\delta_{i,j}$ on the subspace $\ker(P_i[\phi])$ (since $Q_j\equiv \delta_{i,j}\pmod{P_i}$), its image is contained in $\ker(P_j[\phi])$ (since $P_jQ_j\equiv0\pmod P$) and $\pi_1+\cdots+\pi_k=1$ (since $Q_1+\cdots+Q_k\equiv1\pmod P$). This implies that the $\pi_j$ are the projections onto the factors of a direct sum decomposition as announced. See also this answer.

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