UTVt

UTVt refers to a matrix factorization in which a real or complex matrix A is written as A = U T V^T, where U and V are orthogonal (unitary in the complex case) and T is upper triangular. When A is rectangular, T is upper trapezoidal to accommodate the shape; when A is square, T is strictly upper triangular in the sense of containing zeros below the main diagonal. The UTV^T form is a structured alternative to the singular value decomposition (SVD) and the QR or URV factorizations.

The decomposition exists for any matrix A and is not unique, since the orthogonal factors can be modified by certain diagonal or block-diagonal transformations without changing the product. In a rank-revealing variant, the diagonal entries or the prominent elements of T provide information about the numerical rank of A. However, unlike the SVD, the diagonal of T does not generally equal the singular values of A; the diagonal entries offer a triangular surrogate rather than an exact spectrum.

Computation of a UTVt factorization uses a sequence of orthogonal transformations, such as Householder reflectors or Givens rotations, applied from the left and right to push A toward an upper triangular (or upper trapezoidal) form. The accumulations of these transformations are stored in U and V, respectively. Specialized algorithms aim to produce a rank-revealing UTVt factorization, distributing rank information across T to aid interpretation and numerical stability.

Applications of UTVt decompositions include low-rank approximations, preconditioning for iterative methods, and efficient problem reformulations in linear systems and eigenvalue computations. It is related to, and sometimes contrasted with, the SVD and other factorization forms, offering a balance between numerical stability, interpretability, and computational structure.