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

$$\begin{array}{l}Av=\sigma u\\ {A}^{H}u=\sigma v,\end{array}$$

where $${A}^{H}$$ is the Hermitian transpose of *A*. The singular vectors *u* and *v* are typically scaled to have a norm of 1. Also, if *u* and *v* are singular vectors of *A*, then *-u* and *-v* are singular vectors of `A`

as well.

The singular values *σ* are always real and nonnegative, even if *A* is complex. 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 obtain the equations

$$\begin{array}{l}AV=U\Sigma \\ {A}^{H}U=V\Sigma .\end{array}$$

Since *U* and *V* are unitary matrices, multiplying the first equation by $${V}^{H}$$ on the right yields the singular value decomposition equation

$$A=U\Sigma {V}^{H}.$$

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 (`m > n`

), then the resulting `m`

-by-`m`

matrix *U* is large. However, most of the columns in *U* 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.

If the matrix `A`

is large and sparse, then using `svd`

to calculate *all* of the singular values and vectors is not always practical. For example, if you need to know just a few of the largest singular values, then calculating all of the singular values of a 5000-by-5000 sparse matrix is a lot of extra work. In cases where only a subset of the singular values and vectors are required, the `svds`

function is preferred over `svd`

.

For a 1000-by-1000 random sparse matrix with a density of about 30%,

n = 1000; A = sprand(n,n,0.3);

the six largest singular values are

S = svds(A) S = 130.2184 16.4358 16.4119 16.3688 16.3242 16.2838

Also, the six smallest singular values are

S = svds(A,6,'smallest') S = 0.0740 0.0574 0.0388 0.0282 0.0131 0.0066

For smaller matrices that can fit in memory as a full matrix, `full(A)`

, using `svd(full(A))`

might still be quicker than `svds`

. However, for truly large and sparse matrices, using `svds`

becomes necessary.