In the special case when is an real square matrix, the matrices and can be chosen to be real matrices too. In that case, "unitary" is the same as "orthogonal". Then, interpreting both unitary matrices as well as the diagonal matrix, summarized here as , as a linear transformation of the space , the matrices and represent rotations or reflection of the space, while formula_5 represents the scaling of each coordinate by the factor . Thus the SVD decomposition breaks down any invertible linear transformation of into a composition of three geometrical transformations: a rotation or reflection (), followed by a coordinate-by-coordinate scaling (formula_5), followed by another rotation or reflection ().