Transcription of [1] Eigenvectors and Eigenvalues - MIT Mathematics
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[1] Eigenvectors and Eigenvalues - MIT Mathematics
: Eigenvalues and Eigenvectors [1] Eigenvectors and Eigenvalues [2]Observations about Eigenvalues [3]Complete Solution to system of ODEs[4]Computing Eigenvectors [5]Computing Eigenvalues [1] Eigenvectors and EigenvaluesExample from Differential EquationsConsider the system of first order, linear 5y1+ 2y2dy2dt= 2y1+ 5y2We can write this using the companion matrix form:[y 1y 2]=[5 22 5][y1y2].Note that this matrix is symmetric. Using notation from linear algebra,we can write this even more succinctly asy = is a coupled equation, and we want to uncouple of OptimismWe ve seen that solutions to linear ODEs have the formert. So we will lookfor solutions1y1=e tay2=e tbWriting in vector notation:y=e t[ab]=e txHere is the eigenvalue andxis the find a solution of this form, we simply plug in this solution into theequationy =Ay:ddte tx= e txAe tx=e tAxIf there is a solution of this form, it satisfies this equation e tx=e that becausee tis never zero, we can cancel it from both sides ofthis equation, and we end up with the centr
De nitions A nonzero vector x is an eigenvector if there is a number such that Ax = x: The scalar value is called the eigenvalue. ... the diagonal entries of the matrix Ais 5 + 5 = 10. 4. ... De nition The characteristic polynomial of an nby nmatrix Ais the nth degree poly-nomial det(A I). 8 The roots of this polynomial are the eigenvalues of A.
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