06-240/Classnotes For Tuesday December 5

2007-05-28T06:37:17Z

213.185.1.179:

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

'''Theorem.''' Let <math>A</math> be an <math>n\times n</math> matrix (with entries in some field <math>F</math>) and let <math>\chi_A(\lambda):=\det(A-\lambda I)</math> be the characteristic polynomial of <math>A</math>. Assume <math>\chi_A</math> has <math>n</math> distinct roots <math>\lambda_1\ldots\lambda_n</math>, that is, <math>A</math> has <math>n</math> distinct eigenvalues <math>\lambda_1\ldots\lambda_n</math>, and let <math>v_1,\ldots,v_n</math> be corresponding eigenvectors, so that <math>Av_i=\lambda_iv_i</math> for all <math>1\leq i\leq n</math>. Let <math>D</math> be the diagonal matrix that has <math>\lambda_1</math> through <math>\lambda_n</math> on its main diagonal (in order) and let <math>P</math> be the matrix whose columns are these eigenvectors: <math>P:=(v_1|v_2|\cdots|v_n)</math>. Then <math>P</math> is invertible and the following equalities hold:
# <math>D=P^{-1}AP</math> and <math>A=PDP^{-1}</math>.
# For any positive integer <math>k</math> we have <math>A^k=PD^kP^{-1}</math> and <math>D^k=\begin{pmatrix}\lambda_1^k</math>

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06-240/Classnotes For Tuesday December 5