Singular Values

A singular value and corresponding singular vectors of a rectangular matrix A are, respectively, a scalar σ and a pair of vectors u and v that satisfy

Av = σu
Au = σv.

With the singular values on the diagonal of a diagonal matrix Σ and the corresponding singular vectors forming the columns of two orthogonal matrices U and V, you have

AV =
AU = .

Since U and V are orthogonal, this becomes the singular value decomposition

A = UΣV′.

The full singular value decomposition of an m-by-n matrix involves an m-by-m U, an m-by-n Σ, and an n-by-n V. In other words, U and V are both square and Σ is the same size as A. If A has many more rows than columns, the resulting U can be quite large, but most of its columns are multiplied by zeros in Σ. In this situation, the economy sized decomposition saves both time and storage by producing an m-by-n U, an n-by-n Σ and the same V.

The eigenvalue decomposition is the appropriate tool for analyzing a matrix when it represents a mapping from a vector space into itself, as it does for an ordinary differential equation. However, the singular value decomposition is the appropriate tool for analyzing a mapping from one vector space into another vector space, possibly with a different dimension. Most systems of simultaneous linear equations fall into this second category.

If A is square, symmetric, and positive definite, then its eigenvalue and singular value decompositions are the same. But, as A departs from symmetry and positive definiteness, the difference between the two decompositions increases. In particular, the singular value decomposition of a real matrix is always real, but the eigenvalue decomposition of a real, nonsymmetric matrix might be complex.

For the example matrix

A =
     9     4
     6     8
     2     7

the full singular value decomposition is

[U,S,V] = svd(A)

U =

    0.6105   -0.7174    0.3355
    0.6646    0.2336   -0.7098
    0.4308    0.6563    0.6194


S =

   14.9359         0
         0    5.1883
         0         0


V =

    0.6925   -0.7214
    0.7214    0.6925

You can verify that U*S*V' is equal to A to within round-off error. For this small problem, the economy size decomposition is only slightly smaller:

[U,S,V] = svd(A,0)

U =

    0.6105   -0.7174
    0.6646    0.2336
    0.4308    0.6563


S =

   14.9359         0
         0    5.1883


V =

    0.6925   -0.7214
    0.7214    0.6925

Again, U*S*V' is equal to A to within round-off error.

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