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Expressing the eigenvectors of the diagonalized matrix in the basis :
These eigenvalues are the values that will appear in the diagonalized form of matrix A, so by finding the eigenvalues of A we have diagonalized it.
Note there is no preferred order of the eigenvectors in P; changing the order of the eigenvectors in P just changes the order of the eigenvalues in the diagonalized form of A.
Wörterbuch für Mathematik