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== Operator matrix in eigenbasis ==
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▲{{Main|Transformation_matrix#Finding the matrix of a transformation|Eigenvalues and eigenvectors|l1=Finding the matrix of a transformation}}
As explained in [[transformation matrix#Finding the matrix of a transformation|determining coefficients of operator matrix]], there is a special basis, '''e'''<sub>1</sub>, …, '''e'''<sub>''n''</sub>, for which the matrix <math>A</math> takes the diagonal form. Hence, in the defining equation <math display="inline">A \mathbf e_j = \sum a_{i,j} \mathbf e_i</math>, all coefficients <math>a_{i,j} </math> with ''i'' ≠ ''j'' are zero, leaving only one term per sum. The surviving diagonal elements, <math>a_{i,i}</math>, are known as '''eigenvalues''' and designated with <math>\lambda_i</math> in the equation, which reduces to <math>A \mathbf e_i = \lambda_i \mathbf e_i</math>. The resulting equation is known as '''eigenvalue equation'''<ref>{{cite book |last=Nearing |first=James |year=2010 |title=Mathematical Tools for Physics |url=http://www.physics.miami.edu/nearing/mathmethods |chapter=Chapter 7.9: Eigenvalues and Eigenvectors |chapter-url= http://www.physics.miami.edu/~nearing/mathmethods/operators.pdf |access-date=January 1, 2012|isbn=048648212X}}</ref> and used to derive the [[characteristic polynomial]] and, further, [[eigenvalues and eigenvectors]].
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