LSI [BDO95] exploits the factorization of the
matrix **A** into the product of **3** matrices
using the singular value decomposition (SVD).
Given an matrix **A**, where and
rank(**A**) = **r**, the singular value decomposition of **A** is defined as

where and . The first **r** columns of the orthogonal matrices
**U** and **V** define the orthonormal eigenvectors associated with the **r** nonzero
eigenvalues of and , respectively. The columns of
**U** and **V** are referred to as the left and right singular vectors,
respectively, and
the singular values of A are the diagonal elements of or
the nonnegative square roots of the **n** eigenvalues of
[GL89].

As defined by Equation (3), the SVD
is used to represent
the original relationships among terms and documents as sets of
linearly-independent vectors or * factor values*. Using
**k** factors or the **k**-largest singular values and corresponding
singular vectors one can encode (see [BDO95])
the original
term-by-document matrix as a smaller (and more reliable) collection of vectors
in **k**-space for conceptual query processing.

Michael W. Berry (berry@cs.utk.edu)

Mon Jan 29 14:30:24 EST 1996