Editing Matrix decomposition (section)

=== Eigendecomposition ===
{{main|Eigendecomposition (matrix)}}
*Also called ''[[Spectral decomposition (Matrix)|spectral decomposition]]''.
*Applicable to: [[square matrix]] ''A'' with linearly independent eigenvectors (not necessarily distinct eigenvalues).
*Decomposition: <math>A=VDV^{-1}</math>, where ''D'' is a [[diagonal matrix]] formed from the [[eigenvalue]]s of ''A'', and the columns of ''V'' are the corresponding [[eigenvector]]s of ''A''.
*Existence: An ''n''-by-''n'' matrix ''A'' always has ''n'' (complex) eigenvalues, which can be ordered (in more than one way) to form an ''n''-by-''n'' diagonal matrix ''D'' and a corresponding matrix of nonzero columns ''V'' that satisfies the [[eigenvalue equation]] <math>AV=VD</math>.  <math>V</math> is invertible if and only if the ''n'' eigenvectors are [[Linear independence|linearly independent]] (that is, each eigenvalue has [[geometric multiplicity]] equal to its [[algebraic multiplicity]]). A sufficient (but not necessary) condition for this to happen is that all the eigenvalues are different (in this case geometric and algebraic multiplicity are equal to 1)
*Comment: One can always normalize the eigenvectors to have length one (see the definition of the eigenvalue equation)
*Comment: Every [[normal matrix]] ''A'' (that is, matrix for which <math>AA^*=A^*A</math>, where <math>A^*</math> is a [[conjugate transpose]]) can be eigendecomposed. For a [[normal matrix]] ''A'' (and only for a normal matrix), the eigenvectors can also be made orthonormal (<math>VV^*=I</math>) and the eigendecomposition reads as <math>A=VDV^*</math>. In particular all [[Unitary matrix|unitary]], [[Hermitian matrix|Hermitian]], or [[Skew-Hermitian matrix|skew-Hermitian]] (in the real-valued case, all [[Orthogonal matrix|orthogonal]], [[Symmetric matrix|symmetric]], or [[Skew-symmetric matrix|skew-symmetric]], respectively) matrices are normal and therefore possess this property.
*Comment: For any real [[symmetric matrix]] ''A'', the eigendecomposition always exists and can be written as <math>A=VDV^\mathsf{T}</math>, where both ''D'' and ''V'' are real-valued.
*Comment: The eigendecomposition is useful for understanding the solution of a system of linear ordinary differential equations or linear difference equations. For example, the difference equation <math>x_{t+1}=Ax_t</math> starting from the initial condition <math>x_0=c</math> is solved by <math>x_t = A^tc</math>, which is equivalent to <math>x_t = VD^tV^{-1}c</math>, where ''V'' and ''D'' are the matrices formed from the eigenvectors and eigenvalues of ''A''. Since ''D'' is diagonal, raising it to power <math>D^t</math>, just involves raising each element on the diagonal to the power ''t''. This is much easier to do and understand than raising ''A'' to power ''t'', since ''A'' is usually not diagonal.