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{{Main|Transformation matrix#Finding the matrix of a transformation|Eigenvalues and eigenvectors}}
As explained in [[transformation matrix#Finding the matrix of a transformation|determining coefficients of operator matrix]], there is a special basis, {{math|'''e'''<sub>1</sub>, ..., '''e'''<sub>''n''</sub>}}, for which the matrix <math>\mathbf{A}</math> takes the diagonal form. Hence, in the defining equation <math display="inline">\mathbf{A} \mathbf e_j = \sum_i a_{i,j} \mathbf e_i</math>, all coefficients <math>a_{i,j} </math> with {{math|''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>\mathbf{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=978-0486482125}}</ref> and used to derive the [[characteristic polynomial]] and, further, [[eigenvalues and eigenvectors]].
In other words, the [[eigenvalue]]s of {{math|diag(''λ''<sub>1</sub>, ..., ''λ''<sub>''n''</sub>)}} are {{math|''λ''<sub>1</sub>, ..., ''λ''<sub>''n''</sub>}} with associated [[eigenvectors]] of {{math|'''e'''<sub>1</sub>, ..., '''e'''<sub>''n''</sub>}}.
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