Editing Singular value decomposition (section)

=== The columns of {{math|U}} and {{math|V}} are orthonormal bases ===
Since {{tmath|\mathbf U}} and {{tmath|\mathbf V^*}} are unitary, the columns of each of them form a set of [[orthonormal vectors]], which can be regarded as [[basis vectors]]. The matrix {{tmath|\mathbf M}} maps the basis vector {{tmath|\mathbf V_i}} to the stretched unit vector {{tmath|\sigma_i \mathbf U_i.}}  By the definition of a unitary matrix, the same is true for their conjugate transposes {{tmath|\mathbf U^*}} and {{tmath|\mathbf V,}} except the geometric interpretation of the singular values as stretches is lost. In short, the columns of {{tmath|\mathbf U,}} {{tmath|\mathbf U^*,}} {{tmath|\mathbf V,}} and {{tmath|\mathbf V^*}} are [[Orthonormal basis|orthonormal bases]]. When {{tmath|\mathbf M}} is a [[Definite matrix|positive-semidefinite]] [[Hermitian matrix]], {{tmath|\mathbf U}} and {{tmath|\mathbf V}} are both equal to the unitary matrix used to diagonalize  {{tmath|\mathbf M.}} However, when {{tmath|\mathbf M}} is not positive-semidefinite and Hermitian but still [[diagonalizable]], its [[eigendecomposition]] and singular value decomposition are distinct.