Singular Value Decomposition - Singular Values, Singular Vectors, and Their Relation To The SVD

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.