Editing Invertible matrix (section)

=== Eigendecomposition ===
{{Main article|Eigendecomposition of a matrix}}
If matrix {{math|'''A'''}} can be eigendecomposed, and if none of its eigenvalues are zero, then {{math|'''A'''}} is invertible and its inverse is given by
: <math>\mathbf{A}^{-1} = \mathbf{Q}\mathbf{\Lambda}^{-1}\mathbf{Q}^{-1},</math>

where {{math|'''Q'''}} is the square {{math|(''N'' × ''N'')}} matrix whose {{mvar|i}}th column is the [[eigenvector]] <math>q_i</math> of {{math|'''A'''}}, and {{math|'''Λ'''}} is the [[diagonal matrix]] whose diagonal entries are the corresponding eigenvalues, that is, <math>\Lambda_{ii} = \lambda_i.</math> If
{{math|'''A'''}} is symmetric, {{math|'''Q'''}} is guaranteed to be an [[orthogonal matrix]], therefore <math>\mathbf{Q}^{-1} = \mathbf{Q}^\mathrm{T} .</math> Furthermore, because {{math|'''Λ'''}} is a diagonal matrix, its inverse is easy to calculate:
: <math>\left[\Lambda^{-1}\right]_{ii} = \frac{1}{\lambda_i}.</math>