Theorems about matrices, characteristic polynomials, minimal polynomials

One or two of you missed class on Friday (well, four of you, actually). I gave a particularly riveting lecture, in which I gave a summary of what we've done and outline of what's coming next. First, here's the summary:

• Setup: T is a linear operator on a finite-dimensional vector space V. V has dimension n. T has eigenvalues c₁, ..., c_k, with corresponding eigenspaces W₁, ..., W_k. Write f(x) for the characteristic polynomial for T; write p(x) for the minimal polynomial.
• Theorem: T is diagonalizable if and only if the characteristic polynomial for T is

     (x-c1)d1 (x-c2)d2 ... (x-ck)dk
  

  where each d_i is the dimension of the corresponding W_i.
• Theorem (continued): T is diagonalizable if and only if

     dim W1 + dim W2 + ... + dim Wk = n.
  

• Example: If T has n distinct eigenvalues, where n = dim V, then T is diagonalizable.
• Theorem: p(x) and f(x) have the same roots (but not necessarily with the same multiplicities).

• Theorem (Cayley-Hamilton): f(T)=0; hence p(x) divides evenly into f(x).
• Definition: T is triangulable if there is a basis for V with respect to which T is represented by an upper triangular matrix.
• Theorem: T is triangulable if and only if the minimal polynomial p(x) factors into linear terms.
• Corollary: If we're working over the complex numbers, then every T is triangulable.
• Theorem: T is diagonalizable if and only if the minimal polynomial p(x) factors into distinct linear terms.

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