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wwm
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So my professor gave me an extra problem for Linear Algebra and I can't find anything about it in his lecture notes or textbooks or online. I think I've made it through some of the more difficult stuff, but I am running into a catch at the end.
Find [;T(p(x))^{500};] when [;T(p(x))= p(x)-2p'(x)+3p''(x);] and [;p(x)=\alpha+\beta x+\gamma x^{2};] where [;p'(x);] is the first derivative, etc.
A is a matrix representing the transformation. [;T(y)^{500}=A^{500}y;]
Normally we would simply find A by plugging in the std. basis and then find a diagonal matrix, D, similar to A s.t. [;A^{500}=SD^{500}S^{-1};] with D constructed out of the eigenvalues and the transition matrix, S, constructed out of the corresponding eigenvectors.
The trick is that this matrix (and all upper or lower triangle matrices) are not properly diagonizable. We find this out in the last step mentioned because the dimension of the eigenvectors (in this case, 2) do not equal the multiplicity the eigenvalues (in this case 3).
Our professor says, then, that we must add a nilpotent matrix to our equation to get [;A^{500}=S(D+N)^{500}S^{-1};] where N is the nilpotent matrix made of the non-diagonal numbers from our original matrix A. Because the nilpotent matrix goes to zero after several powers, the [;(D+N)^{500};] can be reduced to only the first several terms, which is still a bit hairy, but not too bad. so we finally get our matrix [;(D+N)^{500};], great, hard part over.
2. Relevant questions
(1) But now that we are ready to do our calculations I realize that I don't have enough eigenvectors to create S and thus its inverse as well. Where do I get the last eigenvector?
(1a.) A friend suggested that I don't need the transition matrices since in this case the diagonals in A are the eigenvalues. Is that correct? More importantly, if that wasn't the case where would I get them (I expect a similar problem on my final)?
(1b.)My professor said I need to solve the non-homogenous system using the previous eigenvectors, but I am not sure I fully understand it. What I copied down from him was [;(A-\lambda I)u1=0;] and [;(A-\lambda I)u2=u1;] where u1 is one of the original eigenvectors and u2 is our new one. However, when I go through this process I get the same two eigenvectors, no new third one. What am I doing wrong?
(2) Does the method I described above to tackle the problem make sense? I am going about it correctly?
Homework Statement
Find [;T(p(x))^{500};] when [;T(p(x))= p(x)-2p'(x)+3p''(x);] and [;p(x)=\alpha+\beta x+\gamma x^{2};] where [;p'(x);] is the first derivative, etc.
The Attempt at a Solution
A is a matrix representing the transformation. [;T(y)^{500}=A^{500}y;]
Normally we would simply find A by plugging in the std. basis and then find a diagonal matrix, D, similar to A s.t. [;A^{500}=SD^{500}S^{-1};] with D constructed out of the eigenvalues and the transition matrix, S, constructed out of the corresponding eigenvectors.
The trick is that this matrix (and all upper or lower triangle matrices) are not properly diagonizable. We find this out in the last step mentioned because the dimension of the eigenvectors (in this case, 2) do not equal the multiplicity the eigenvalues (in this case 3).
Our professor says, then, that we must add a nilpotent matrix to our equation to get [;A^{500}=S(D+N)^{500}S^{-1};] where N is the nilpotent matrix made of the non-diagonal numbers from our original matrix A. Because the nilpotent matrix goes to zero after several powers, the [;(D+N)^{500};] can be reduced to only the first several terms, which is still a bit hairy, but not too bad. so we finally get our matrix [;(D+N)^{500};], great, hard part over.
2. Relevant questions
(1) But now that we are ready to do our calculations I realize that I don't have enough eigenvectors to create S and thus its inverse as well. Where do I get the last eigenvector?
(1a.) A friend suggested that I don't need the transition matrices since in this case the diagonals in A are the eigenvalues. Is that correct? More importantly, if that wasn't the case where would I get them (I expect a similar problem on my final)?
(1b.)My professor said I need to solve the non-homogenous system using the previous eigenvectors, but I am not sure I fully understand it. What I copied down from him was [;(A-\lambda I)u1=0;] and [;(A-\lambda I)u2=u1;] where u1 is one of the original eigenvectors and u2 is our new one. However, when I go through this process I get the same two eigenvectors, no new third one. What am I doing wrong?
(2) Does the method I described above to tackle the problem make sense? I am going about it correctly?
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