## 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

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

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 in 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:

*m*-by-*m*matrix*U**m*-by-*n*matrix*Σ**n*-by-*n*matrix*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,"econ") 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.

### Batched SVD Calculation

If you need to decompose a large collection of matrices that have the same size, it is
inefficient to perform all of the decompositions in a loop with
`svd`

. Instead, you can concatenate all of the matrices into a
multidimensional array and use `pagesvd`

to perform singular value decompositions on all of the array pages with a single
function call.

Function | Usage |
---|---|

`pagesvd` | Use `pagesvd` to perform singular value
decompositions on the pages of a multidimensional array. This is an
efficient way to perform SVD on a large collection of matrices that all
have the same size. |

For example, consider a collection of three 2-by-2 matrices. Concatenate the matrices
into a 2-by-2-by-3 array with the `cat`

function.

A = [0 -1; 1 0]; B = [-1 0; 0 -1]; C = [0 1; -1 0]; X = cat(3,A,B,C);

Now, use `pagesvd`

to simultaneously perform the three
decompositions.

[U,S,V] = pagesvd(X);

For each page of `X`

, there are corresponding pages in the outputs
`U`

, `S`

, and `V`

. For example,
the matrix `A`

is on the first page of `X`

, and its
decomposition is given by `U(:,:,1)*S(:,:,1)*V(:,:,1)'`

.

### Low-Rank SVD Approximations

For large sparse matrices, using `svd`

to calculate
*all* of the singular values and singular 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 extra
work.

In cases where only a subset of the singular values and singular vectors are required,
the `svds`

and `svdsketch`

functions are preferred
over `svd`

.

Function | Usage |
---|---|

`svds` | Use `svds` to calculate a
rank-k approximation of the SVD. You can
specify whether the subset of singular values should be the largest,
the smallest, or the closest to a specific number.
`svds` generally calculates the best possible
rank-k approximation. |

`svdsketch` | Use `svdsketch` to calculate a partial SVD of
the input matrix satisfying a specified tolerance. While
`svds` requires that you specify the rank,
`svdsketch` adaptively determines the rank of
the matrix sketch based on the specified tolerance. The
rank-k approximation that
`svdsketch` ultimately uses satisfies the
tolerance, but unlike `svds` , it is not
guaranteed to be the best one possible. |

For example, consider 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`

or `svdsketch`

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

or
`svdsketch`

becomes necessary.

## See Also

`svd`

| `svds`

| `svdsketch`

| `gsvd`

| `pagesvd`