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 A be an n\times n matrix (with entries in some field F) and let \chi_A(\lambda):=\det(A-\lambda I) be the characteristic polynomial of A. Assume \chi_A has n distinct roots \lambda_1\ldots\lambda_n, that is, A has n distinct eigenvalues \lambda_1\ldots\lambda_n, and let v_1,\ldots,v_n be corresponding eigenvectors, so that Av_i=\lambda_iv_i for all 1\leq i\leq n. Let D be the diagonal matrix that has \lambda_1 through \lambda_n on its main diagonal (in order) and let P be the matrix whose columns are these eigenvectors: P:=(v_1|v_2|\cdots|v_n). Then P is invertible and the following equalities hold:

  1. D=P^{-1}AP and A=PDP^{-1}.
  2. For any positive integer k we have A^k=PD^kP^{-1} and D^k=\begin{pmatrix}\lambda_1^k&&0\\&\ddots&\\0&&\lambda_n^k\end{pmatrix}.
  3. Likewise if F={\mathbb R} and \exp(B):=\sum_{k=0}^\infty\frac{B^k}{k!} then \exp(A)=P\exp(D)P^{-1} and \exp(D)=\begin{pmatrix}e^{\lambda_1}&&0\\&\ddots&\\0&&e^{\lambda_n}\end{pmatrix}.

Order of the proceedings.

  1. Assuming P is invertible, a proof of 1.
  2. Proof of 2.
  3. Example - the "reproduction of rabbits" matrix A=\begin{pmatrix}0&1\\1&1\end{pmatrix} (see the mathematica session below).
  4. Discussion of 3.
  5. The relationship with linear transformations and changes of basis.
  6. v_1 thorough v_n form a basis and P is invertible.

06-240-Reproduction of Rabbits.png