Editing Singular value decomposition (section)

=== Rotation, coordinate scaling, and reflection ===
In the special case when {{tmath|\mathbf M}} is an {{tmath|m \times m}} real [[square matrix]], the matrices {{tmath|\mathbf U}} and {{tmath|\mathbf V^*}} can be chosen to be real {{tmath|m \times m}} matrices too.  In that case, "unitary" is the same as "[[orthogonal matrix|orthogonal]]".  Then, interpreting both unitary matrices as well as the diagonal matrix, summarized here as {{tmath|\mathbf A,}} as a [[linear transformation]] {{tmath| \mathbf x \mapsto \mathbf{Ax} }} of the space {{tmath|\mathbf R_m,}} the matrices {{tmath|\mathbf U}} and {{tmath|\mathbf V^*}} represent [[rotation (geometry)|rotations]] or [[reflection (geometry)|reflection]] of the space, while {{tmath|\mathbf \Sigma}} represents the [[scaling matrix|scaling]] of each coordinate {{tmath|\mathbf x_i}} by the factor {{tmath|\sigma_i.}}  Thus the SVD decomposition breaks down any linear transformation of {{tmath|\mathbf R^m}} into a [[function composition|composition]] of three geometrical [[transformation (geometry)|transformations]]: a rotation or reflection {{nobr|({{tmath|\mathbf V^*}}),}} followed by a coordinate-by-coordinate [[scaling (geometry)|scaling]] {{nobr|({{tmath|\mathbf \Sigma}}),}} followed by another rotation or reflection {{nobr|({{tmath|\mathbf U}}).}}

In particular, if {{tmath|\mathbf M}} has a positive determinant, then {{tmath|\mathbf U}} and {{tmath|\mathbf V^*}} can be chosen to be both rotations with reflections, or both rotations without reflections.{{Citation needed|date=September 2022}} If the determinant is negative, exactly one of them will have a reflection. If the determinant is zero, each can be independently chosen to be of either type.

If the matrix {{tmath|\mathbf M}} is real but not square, namely {{tmath|m\times n}} with {{tmath|m \neq n,}} it can be interpreted as a linear transformation from {{tmath|\mathbf R^n}} to  {{tmath|\mathbf R^ m.}}  Then {{tmath|\mathbf U}} and {{tmath|\mathbf V^*}} can be chosen to be rotations/reflections of {{tmath|\mathbf R^m}} and {{tmath|\mathbf R^n,}} respectively; and {{tmath|\mathbf \Sigma,}} besides scaling the first {{tmath|\min\{m,n\} }} coordinates, also extends the vector with zeros, i.e. removes trailing coordinates, so as to turn {{tmath|\mathbf R^n}} into {{tmath|\mathbf R^m.}}