Editing Transformation matrix (section)

===Eigenbasis and diagonal matrix===
{{Main|Diagonal matrix|Eigenvalues and eigenvectors}}

Yet, there is a special basis for an operator in which the components form a [[diagonal matrix]] and, thus, multiplication complexity reduces to {{mvar|n}}. Being diagonal means that all coefficients <math>a_{i,j} </math> except <math>a_{i,i}</math> are zeros leaving only one term in the sum <math display="inline">\sum a_{i,j} \mathbf e_i</math> above. The surviving diagonal elements, <math>a_{i,i}</math>, are known as '''eigenvalues''' and designated with <math>\lambda_i</math> in the defining 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 = 978-0486482125 }}</ref> The [[Eigenvalues and eigenvectors|eigenvectors and eigenvalues are derived from it via the '''characteristic polynomial''']].

With [[Diagonalizable matrix#Diagonalization|diagonalization]], it is [[diagonalizability|often possible]] to [[change of basis|translate]] to and from eigenbases.