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Let $A_1,\cdots, A_n$ be $n\times n$ matrices. If $A_i^2=A_i$, $A_iA_j=0$ for $i\neq j$. Show that there exists an invertible matrix $P$ such that $P^{-1}A_iP$ are all diagonal..

I only know that all $A_i$ can be diagonalized. However, this same $P$ I could not find...

xldd
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2 Answers2

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You can read theorem 5.1 in this great paper by K. Conrad , stating that two or more diagonalizable linear maps are simultaneously diagonalizable iff they commute.

In our case, each map is a zero of the polynomial

$$x^2-x=x(x-1)$$

and then the minimal map of each map divides the above polynomial and is thus a product of different linear maps, which means each map is diagonalizable (you can read about this also in that paper), and the condition $\;A_iA_j=0\;\;\forall\;i\neq j\;$ gives us pairwise commutativity, and thus we're done.

Timbuc
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Proof by induction on $n$:

For $n=1$, its obvious;

Suppose its right for $n-1$, then for $n$, there exist an invertible matrix $P$ such that $P^{-1}A_1P=\left( \begin{array}{cc} D_1& \\ & 0 \\ \end{array} \right)$ ,where $D_1$ is a diagnol invertible matrix, $P^{-1}A_1PP^{-1}A_i P=P^{-1}A_i PP^{-1}A_1 P=0$ ,so $P^{-1}A_i P=\left( \begin{array}{cc} 0& \\ & D_i \\ \end{array} \right)$ ,where $D_i$ is an low dim matrix and $D_i^2=D_i$ and $D_i D_j=D_j D_i$, so there exist some matrix $Q$ such that $Q^{-1}D_iQ$ are all diagonal, then set $R=P\left( \begin{array}{cc} I& \\ & Q \\ \end{array} \right)$, $R^{-1}A_i R$ are all diagonal.

Alfred Chern
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