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from which it follows that there is no g € G that will make g-A upper or lower triangular, so A is not triangularizable. Also, by Lemmaf[2.1] none of the 3 commutators between the pairs will vanish, so A is reduced.

Figure 1 from which it follows that there is no g € G that will make g-A upper or lower triangular, so A is not triangularizable. Also, by Lemmaf[2.1] none of the 3 commutators between the pairs will vanish, so A is reduced.

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Theorem 4.3. All non-commutative matrix sequences can be put in canonical form. More precisely, after rearranging terms, any sequence is similar to a sequence with A,, Ao and Az as described by the following table. Proof. Let us show that only the five possibilities above occur, and may be put in the given forms, starting when A is stable. If n > 4 or n = 2, Proposition [4.2] implies that there is some 0 # 0, so we can rearrange the terms of A so that we have oj2 4 O (in particular, [A,, Ag] 4 0). If A; or Ag is diagonalizable, we have possibility 1.a. Assuming Proof. If A is not stable, then it is triangularizable, by Proposition B.5] so that all o = 0, by Proposition 2.10} Conversely, suppose A is stable with reduced length n = 2. Then o12 # 0 because of Friedland’s result (or Theorem [L11). Finally, if n > 4, Proposition 2.20) implies that at least one oj, #0. On the other hand, Example [2.11] shows that the statement is not true for n = 3.
Theorem 4.3. All non-commutative matrix sequences can be put in canonical form. More precisely, after rearranging terms, any sequence is similar to a sequence with A,, Ao and Az as described by the following table. Proof. Let us show that only the five possibilities above occur, and may be put in the given forms, starting when A is stable. If n > 4 or n = 2, Proposition [4.2] implies that there is some 0 # 0, so we can rearrange the terms of A so that we have oj2 4 O (in particular, [A,, Ag] 4 0). If A; or Ag is diagonalizable, we have possibility 1.a. Assuming Proof. If A is not stable, then it is triangularizable, by Proposition B.5] so that all o = 0, by Proposition 2.10} Conversely, suppose A is stable with reduced length n = 2. Then o12 # 0 because of Friedland’s result (or Theorem [L11). Finally, if n > 4, Proposition 2.20) implies that at least one oj, #0. On the other hand, Example [2.11] shows that the statement is not true for n = 3.
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Algebraic GeometryInvariant Theory
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