By Brian Osserman
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Additional resources for A Galois theory example
Wm . We claim that in fact Wi = E∞ (λi ), for each i, which completes the proof. 17. ) To see this claim, let Ti be the restriction of T = TA to Wi . Then Bi = [Ti ]Bi . 16, (Bi − λi I )ki = 0. , Wi ⊆ E∞ (λi ). On the other hand, if v in V is not in Wi , then, writing v = w1 + . . + wm , with wj in Wj for each j , there must be some value of j = i with wj = 0. But (Bj − λi I ) is a block upper triangular matrix with constant diagonal λj − λi = 0, so in particular (Bj − λi I ) is nonsingular, as is any of its powers.
Proof. Let V have dimension n. We prove this by complete induction on n. If n = 1, then the only possible linear transformation T is T (v) = λv for some λ, and then, in any basis B , [T ]B is the 1-by-1 matrix [λ], which is certainly in Jordan Canonical Form. Now for the inductive step. In the course of this argument, we will sometimes be explicitly changing bases, and sometimes conjugating by nonsingular matrices. Of course, these have the same effect (cf. 12), but sometimes one will be more convenient than the other.
6) Let A be an n-by-n upper triangular matrix with distinct diagonal entries d1 , . , dn . 6, for any polynomial f (x), f (A) is an upper triangular matrix with entries f (d1 ), f (d2 ), …, f (dn ). Thus, if f (x) is any polynomial with f (A) = 0, we must have f (d1 ) = . . = f (dn ) = 0, so f (x) must be divisible by the polynomial f0 (x) = (x − d1 ) · · · (x − dn ). On the other hand, a very unenlightening computation shows that f0 (x) = 0. , mA (x) = (x − d1 ) · · · (x − dn ). Also, cA (x) = (x − d1 ) · · · (x − dn ).
A Galois theory example by Brian Osserman