Summary: We introduce a new orthogonalization method. Given a real \(m \times n\) matrix \(A\), the new method constructs an SVD-type decomposition of the form \(A = {\hat U }{\hat \Sigma }{\hat V}^{T}\). The columns of \(\hat U\) and \(\hat V\) are orthonormal, or nearly orthonormal, while \(\hat\Sigma\) is a diagonal matrix whose diagonal entries approximate the singular values of \(A\). The method has three versions: a “left-side” orthogonalization scheme in which the columns of \(\hat U\) constitute an orthonormal basis of Range\((A)\), a “right-side” orthogonalization scheme in which the columns of \(\hat V\) constitute an orthonormal basis of Range\((A^T)\), and a third version in which both \(\hat U\) and \(\hat V\) have orthonormal columns, but the decomposition is not exact. The new decompositions may substitute the SVD in many applications.