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