Singular Values, Singular Vectors, and Their Relation To The SVD
A non-negative real number σ is a singular value for M if and only if there exist unit-length vectors u in Km and v in Kn such that
The vectors u and v are called left-singular and right-singular vectors for σ, respectively.
In any singular value decomposition
the diagonal entries of Σ are equal to the singular values of M. The columns of U and V are, respectively, left- and right-singular vectors for the corresponding singular values. Consequently, the above theorem implies that:
- An m × n matrix M has at least one and at most p = min(m,n) distinct singular values.
- It is always possible to find an orthogonal basis U for Km consisting of left-singular vectors of M.
- It is always possible to find an orthogonal basis V for Kn consisting of right-singular vectors of M.
A singular value for which we can find two left (or right) singular vectors that are linearly independent is called degenerate.
Non-degenerate singular values always have unique left- and right-singular vectors, up to multiplication by a unit-phase factor eiφ (for the real case up to sign). Consequently, if all singular values of M are non-degenerate and non-zero, then its singular value decomposition is unique, up to multiplication of a column of U by a unit-phase factor and simultaneous multiplication of the corresponding column of V by the same unit-phase factor.
Degenerate singular values, by definition, have non-unique singular vectors. Furthermore, if u1 and u2 are two left-singular vectors which both correspond to the singular value σ, then any normalized linear combination of the two vectors is also a left-singular vector corresponding to the singular value σ. The similar statement is true for right-singular vectors. Consequently, if M has degenerate singular values, then its singular value decomposition is not unique.
Read more about this topic: Singular Value Decomposition
Famous quotes containing the words singular and/or relation:
“though the fall cold
surrounds our warm bed, and though
by day we are singular and often lonely.”
—Denise Levertov (b. 1923)
“There is a constant in the average American imagination and taste, for which the past must be preserved and celebrated in full-scale authentic copy; a philosophy of immortality as duplication. It dominates the relation with the self, with the past, not infrequently with the present, always with History and, even, with the European tradition.”
—Umberto Eco (b. 1932)