 My Math Forum Finding eigenvectors without knowing the matrix?

 Linear Algebra Linear Algebra Math Forum

 December 16th, 2017, 02:45 AM #1 Member   Joined: Apr 2014 From: Greece Posts: 58 Thanks: 0 Finding eigenvectors without knowing the matrix? Let $\displaystyle B=A^{12}-8A^{7}+5A^{5}+4I$,$\displaystyle A=\begin{pmatrix} 2 & 1 & 2\\ 0 & 4 & 1\\ 0 & -2 & 1 \end{pmatrix}$ Find two independent eigenvectors for B. How do I do this? I found that the eigenvalues of A are 3 , 2 (double root) and that the eigenspaces are $\displaystyle V(2)={x_{1}\begin{pmatrix} 1 & 0 & 0 \end{pmatrix}^{t},x_{1}\epsilon\mathbb{R}}$ and $\displaystyle V(3)={x_{1}\begin{pmatrix} 1 & -1 & 1 \end{pmatrix}^{t},x_{1}\epsilon\mathbb{R}}$ December 16th, 2017, 03:26 AM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,664 Thanks: 2644 Math Focus: Mainly analysis and algebra Recall that (subject to certain conditions)$$A=SPS^{-1}$$ where $S$ is formed from the eigenvectors of $A$ and $P$ is the diagonal matrix having the eigenvalues of $A$ as elements in the same order as the corresponding eigenvectors. December 16th, 2017, 04:30 AM   #3
Senior Member

Joined: Aug 2017
From: United Kingdom

Posts: 312
Thanks: 111

Math Focus: Number Theory, Algebraic Geometry
Quote:
 Originally Posted by v8archie Recall that (subject to certain conditions)$$A=SPS^{-1}$$ where $S$ is formed from the eigenvectors of $A$ and $P$ is the diagonal matrix having the eigenvalues of $A$ as elements in the same order as the corresponding eigenvectors.
But neither A nor B here satisfy these conditions (they both have an eigenvalue whose geometric multiplicity is less than its algebraic multiplicity), so they aren't diagonalisable. December 20th, 2017, 04:27 AM #4 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 If v is an eigenvector of A, with eigenvalue $\lambda$ then $Bv= A^{12}v- 8A^7v+ 5A^5v+ 4v= \lambda^{12}v- 8\lambda^7v+ 5\lambda^5v+ 4v= (\lambda^{12}- 8\lambda^7+ 5\lambda^5+ 4)v$. That is, any eigenvector of A, with eigenvalue $\lambda$, is an eigenvector of B with eigenvalue $\lambda^{12}- 8\lambda^7+ 5\lambda^5+ 4$ Tags eigenvectors, finding, knowing, matrix Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post whitegreen Linear Algebra 1 August 17th, 2016 06:55 AM tomiojap Linear Algebra 1 March 20th, 2013 03:02 PM aschenbr.rach Calculus 1 March 7th, 2010 11:52 AM asmatic Linear Algebra 0 July 31st, 2009 10:51 AM jokerthief Algebra 3 November 4th, 2007 02:52 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top      