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 s, then any normalized linear combination of the two vectors is also a left-singular vector corresponding to the singular value s. The similar statement is true for right-singular vectors. Consequently, if M has degenerate singular values, then its singular value decomposition is not unique.

Singular Values and Singular Vectors.

On Feb 22, 2:52 am, "Nasser M. Abbasi" <n. @12000. org> wrote: > On 2/22/2012 1:49 AM, Frank wrote: > > > Hello, > > > We call the singular vectors.

s = svds(A) computes the six largest singular values and associated singular vectors of matrix A. If A is m-by-n, svds(A) manipulates eigenvalues and vectors returned by eigs(B), where B = [sparse(m,m) A; A' sparse(n,n)], to find a few singular values and vectors of A. The positive eigenvalues of the symmetric matrix B are the same as the singular values of A.

Where the columns of U are the left singular vectors (gene coefficient vectors); S (the same dimensions as A) has singular values and is diagonal (mode amplitudes); and VT has rows that are the right singular vectors (expression level vectors). The SVD represents an expansion of the original data in a coordinate system where the covariance matrix is diagonal.

Singular vectors of the linearized equations of motion have been used to study the instability properties of the atmosphere–ocean system and its related predictability. A third use of these singular vectors is proposed here: as part of a strategy to target adaptive observations to “sensitive” parts of the atmosphere. Such observations could be made using unmanned aircraft, though calculations in this paper are motivated by the upstream component of the Fronts and Atlantic Storm-Track Experiment.