06-240/Classnotes For Tuesday December 5

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Our remaining goal for this semester is to study the following theorem:

Theorem. Let be an matrix (with entries in some field ) and let be the characteristic polynomial of . Assume has distinct roots , that is, has distinct eigenvalues , and let be corresponding eigenvectors, so that for all . Let be the diagonal matrix that has through on its main diagonal (in order) and let be the matrix whose columns are these eigenvectors: . Then is invertible and the following equalities hold:

  1. and .
  2. For any positive integer we have and .
  3. Likewise if and then and .

Order of the proceedings.

  1. Assuming P is invertible, a proof of 1.
  2. Proof of 2.
  3. Example - the "reproduction of rabbits" matrix (see the mathematica session below).
  4. Discussion of 3.
  5. The relationship with linear transformations and changes of basis.
  6. thorough form a basis and is invertible.

06-240-Reproduction of Rabbits.png