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Diagonalization

Diagonalization. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB. Diagonalization. If an nxn matrix has n independent eigenvectors (i.e. enough to span ℝ n ), we will be able to perform a “similarity” transformation, to obtain a diagonal

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Diagonalization

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  1. Diagonalization Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  2. Diagonalization If an nxn matrix has n independent eigenvectors (i.e. enough to span ℝn), we will be able to perform a “similarity” transformation, to obtain a diagonal matrix that has the eigenvalues of the original matrix on the diagonal. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  3. Diagonalization If an nxn matrix has n independent eigenvectors (i.e. enough to span ℝn), we will be able to perform a “similarity” transformation, to obtain a diagonal matrix that has the eigenvalues of the original matrix on the diagonal. Here is the procedure: Given an nxn matrix A, find all eigenvalues and eigenvectors, then form a matrix with the eigenvectors as columns (we will call this matrix P). Next find the inverse of P. Now multiply: P-1AP=D, D is the diagonal matrix. We can convert back by multiplying: A=PDP-1. On the following slides are a couple of examples. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  4. Diagonalization Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  5. Diagonalization Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  6. Diagonalization Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  7. Diagonalization Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  8. Diagonalization Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A. Now that we have the eigenvalues and eigenvectors, we form the matrix that has the e-vectors as columns. Call it matrix P. Also find its inverse P-1. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  9. Diagonalization Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A. Now that we have the eigenvalues and eigenvectors, we form the matrix that has the e-vectors as columns. Call it matrix P. Also find its inverse P-1. Use the shortcut for the inverse of a 2x2 matrix Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  10. Diagonalization Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A. Now that we have the eigenvalues and eigenvectors, we form the matrix that has the e-vectors as columns. Call it matrix P. Also find its inverse P-1. Next we multiply P-1AP. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  11. Diagonalization Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A. Now that we have the eigenvalues and eigenvectors, we form the matrix that has the e-vectors as columns. Call it matrix P. Also find its inverse P-1. Next we multiply P-1AP. Notice that matrix AP comes out to be just the eigenvectors times the eigenvalues, as expected. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  12. Diagonalization Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A. Now that we have the eigenvalues and eigenvectors, we form the matrix that has the e-vectors as columns. Call it matrix P. Also find its inverse P-1. Next we multiply P-1AP. We end up with a diagonal matrix that has the eigenvalues on the diagonal (and 0 everywhere else). Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  13. Diagonalization Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  14. Diagonalization Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  15. Diagonalization Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  16. Diagonalization Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  17. Diagonalization Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A. Now that we have the eigenvalues and eigenvectors, we form the matrix that has the e-vectors as columns. Call it matrix P. Also find its inverse P-1. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  18. Diagonalization Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A. Now that we have the eigenvalues and eigenvectors, we form the matrix that has the e-vectors as columns. Call it matrix P. Also find its inverse P-1. Use the shortcut for the inverse of a 2x2 matrix Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  19. Diagonalization Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A. Now that we have the eigenvalues and eigenvectors, we form the matrix that has the e-vectors as columns. Call it matrix P. Also find its inverse P-1. Use the shortcut for the inverse of a 2x2 matrix Next we multiply P-1AP. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  20. Diagonalization Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A. Now that we have the eigenvalues and eigenvectors, we form the matrix that has the e-vectors as columns. Call it matrix P. Also find its inverse P-1. Use the shortcut for the inverse of a 2x2 matrix Next we multiply P-1AP. Notice that matrix AP comes out to be just the eigenvectors times the eigenvalues, as expected. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  21. Diagonalization Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A. Now that we have the eigenvalues and eigenvectors, we form the matrix that has the e-vectors as columns. Call it matrix P. Also find its inverse P-1. Use the shortcut for the inverse of a 2x2 matrix Next we multiply P-1AP. We end up with a diagonal matrix that has the eigenvalues on the diagonal (and 0 everywhere else). Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  22. Diagonalization Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  23. Diagonalization Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  24. Diagonalization Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

  25. Diagonalization Example. Matrix A is given. Find all eigenvalues and their associated eigenvectors. Show how to use these vectors to diagonalize matrix A. Now that we have the eigenvalues and eigenvectors, we form the matrix that has the e-vectors as columns. Call it matrix P. Also find its inverse P-1. Next we multiply P-1AP. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

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