Editing Singular value decomposition (section)

=== Relation to eigenvalue decomposition ===
The singular value decomposition is very general in the sense that it can be applied to any {{tmath|m \times n}} matrix, whereas [[eigenvalue decomposition]] can only be applied to square [[Diagonalizable matrix|diagonalizable matrices]]. Nevertheless, the two decompositions are related.

If {{tmath|\mathbf M}} has SVD {{tmath|\mathbf M  {{=}} \mathbf U \mathbf \Sigma \mathbf V^*,}} the following two relations hold:

<math display=block>\begin{align}
\mathbf{M}^* \mathbf{M} &= \mathbf{V} \mathbf \Sigma^* \mathbf{U}^*\, \mathbf{U} \mathbf \Sigma \mathbf{V}^* = \mathbf{V} (\mathbf \Sigma^* \mathbf \Sigma) \mathbf{V}^*, \\[3mu]
\mathbf{M} \mathbf{M}^* &= \mathbf{U} \mathbf \Sigma \mathbf{V}^*\, \mathbf{V} \mathbf \Sigma^* \mathbf{U}^* = \mathbf{U} (\mathbf \Sigma \mathbf \Sigma^*) \mathbf{U}^*.
\end{align}</math>

The right-hand sides of these relations describe the eigenvalue decompositions of the left-hand sides. Consequently:

* The columns of {{tmath|\mathbf V}} (referred to as right-singular vectors) are [[eigenvectors]] of {{tmath|\mathbf M^* \mathbf M.}}
* The columns of {{tmath|\mathbf U}} (referred to as left-singular vectors) are eigenvectors of {{tmath|\mathbf M \mathbf M^*.}}
* The non-zero elements of {{tmath|\mathbf \Sigma}} (non-zero singular values) are the square roots of the non-zero [[eigenvalues]] of {{tmath|\mathbf M^* \mathbf M}} or {{tmath|\mathbf M \mathbf M^*.}}

In the special case of {{tmath|\mathbf M}} being a [[normal matrix]], and thus also square, the [[Spectral theorem#Finite-dimensional case|spectral theorem]] ensures that it can be [[Unitary transform|unitarily]] [[Diagonalizable matrix|diagonalized]] using a basis of [[eigenvector]]s, and thus decomposed as {{tmath|\mathbf M {{=}} \mathbf U\mathbf D\mathbf U^*}} for some unitary matrix {{tmath|\mathbf U}} and diagonal matrix {{tmath|\mathbf D}} with complex elements {{tmath|\sigma_i}} along the diagonal.  When {{tmath|\mathbf M}} is [[Positive-definite matrix|positive semi-definite]], {{tmath|\sigma_i}} will be non-negative real numbers so that the decomposition {{tmath|\mathbf M {{=}} \mathbf U \mathbf D \mathbf U^*}} is also a singular value decomposition.  Otherwise, it can be recast as an SVD by moving the phase {{tmath|e^{i\varphi} }} of each {{tmath|\sigma_i}} to either its corresponding {{tmath|\mathbf V_i}} or {{tmath|\mathbf U_i.}} The natural connection of the SVD to non-normal matrices is through the [[polar decomposition]] theorem:  {{tmath|\mathbf M {{=}} \mathbf S \mathbf R,}} where {{tmath|\mathbf S {{=}} \mathbf U \mathbf\Sigma \mathbf U^*}} is positive semidefinite and normal, and  {{tmath|\mathbf R {{=}} \mathbf U \mathbf V^*}} is unitary.

Thus, except for positive semi-definite matrices, the eigenvalue decomposition and SVD of {{tmath|\mathbf M,}} while related, differ: the eigenvalue decomposition is {{tmath|1= \mathbf M = \mathbf U \mathbf D \mathbf U^{-1},}} where {{tmath|\mathbf U}} is not necessarily unitary and {{tmath|\mathbf D}} is not necessarily positive semi-definite, while the SVD is {{tmath|1= \mathbf M = \mathbf U \mathbf \Sigma \mathbf V^*,}} where {{tmath|\mathbf \Sigma}} is diagonal and positive semi-definite, and {{tmath|\mathbf U}} and {{tmath|\mathbf V}} are unitary matrices that are not necessarily related except through the matrix {{tmath|\mathbf M.}}  While only [[defective matrix|non-defective]] square matrices have an eigenvalue decomposition, any {{tmath|m \times n}} matrix has a SVD.