# Documentation

### This is machine translation

Translated by
Mouseover text to see original. Click the button below to return to the English verison of the page.

# svd

Singular value decomposition

## Description

example

s = svd(A) returns the singular values of matrix A in descending order.

example

[U,S,V] = svd(A) performs a singular value decomposition of matrix A, such that A = U*S*V'.

example

[U,S,V] = svd(A,'econ') produces an economy-size decomposition of m-by-n matrix A:

• m > n — Only the first n columns of U are computed, and S is n-by-n.

• m = nsvd(A,'econ') is equivalent to svd(A).

• m < n — Only the first m columns of V are computed, and S is m-by-m.

example

[U,S,V] = svd(A,0) produces a different economy-size decomposition of m-by-n matrix A:

• m > nsvd(A,0) is equivalent to svd(A,'econ').

• m <= nsvd(A,0) is equivalent to svd(A).

## Examples

collapse all

Compute the singular values of a full rank matrix.

A = [1 0 1; -1 -2 0; 0 1 -1]
A =

1     0     1
-1    -2     0
0     1    -1

s = svd(A)
s =

2.4605
1.6996
0.2391

Find the singular value decomposition of a rectangular matrix A.

A = [1 2; 3 4; 5 6; 7 8]
A =

1     2
3     4
5     6
7     8

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

-0.1525   -0.8226   -0.3945   -0.3800
-0.3499   -0.4214    0.2428    0.8007
-0.5474   -0.0201    0.6979   -0.4614
-0.7448    0.3812   -0.5462    0.0407

S =

14.2691         0
0    0.6268
0         0
0         0

V =

-0.6414    0.7672
-0.7672   -0.6414

Confirm the relation A = U*S*V', within machine precision.

U*S*V'
ans =

1.0000    2.0000
3.0000    4.0000
5.0000    6.0000
7.0000    8.0000

Find the economy-size decomposition of a rectangular matrix.

A = [1 2; 3 4; 5 6; 7 8]
A =

1     2
3     4
5     6
7     8

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

-0.1525   -0.8226
-0.3499   -0.4214
-0.5474   -0.0201
-0.7448    0.3812

S =

14.2691         0
0    0.6268

V =

-0.6414    0.7672
-0.7672   -0.6414

Since A is 4-by-2, svd(A,'econ') produces the same results.

Use the results of the singular value decomposition to determine the rank, column space, and null space of a matrix.

A = [2 0 2; 0 1 0; 0 0 0]
A =

2     0     2
0     1     0
0     0     0

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

1     0     0
0     1     0
0     0     1

S =

2.8284         0         0
0    1.0000         0
0         0         0

V =

0.7071         0   -0.7071
0    1.0000         0
0.7071         0    0.7071

Calculate the rank using the number of nonzero singular values.

s = diag(S);
rank_A = nnz(s)
rank_A =

2

Compute an orthonormal basis for the column space of A using the columns of U that correspond to nonzero singular values.

column_basis = U(:,logical(s))
column_basis =

1     0
0     1
0     0

Compute an orthonormal basis for the null space of A using the columns of V that correspond to singular values equal to zero.

null_basis = V(:,~s)
null_basis =

-0.7071
0
0.7071

The functions rank, orth, and null provide convenient ways to calculate these quantities.

## Input Arguments

collapse all

Input matrix. A can be either square or rectangular in size.

Data Types: single | double
Complex Number Support: Yes

## Output Arguments

collapse all

Singular values, returned as a column vector. The singular values are nonnegative real numbers listed in decreasing order.

Left singular vectors, returned as the columns of a matrix.

• For an m-by-n matrix A with m > n, the economy-sized decompositions svd(A,'econ') and svd(A,0) compute only the first n columns of U. In this case, the columns of U are orthogonal and U is an m-by-n matrix that satisfies ${U}^{H}U={I}_{n}$.

• For full decompositions, svd(A) returns U as an m-by-m unitary matrix satisfying $U{U}^{H}={U}^{H}U={I}_{m}$. The columns of U that correspond to nonzero singular values form a set of orthonormal basis vectors for the range of A.

Singular values, returned as a diagonal matrix. The diagonal elements of S are nonnegative singular values in decreasing order. The size of S is as follows:

• For an m-by-n matrix A, the economy-sized decomposition svd(A,'econ') returns S as a square matrix of order min([m,n]).

• For full decompositions, svd(A) returns S with the same size as A.

• If m > n, then svd(A,0) returns S as a square matrix of order min([m,n]).

• If m < n, then svd(A,0) returns S with the same size as A.

Right singular vectors, returned as the columns of a matrix.

• For an m-by-n matrix A with m < n, the economy decomposition svd(A,'econ') computes only the first m columns of V. In this case, the columns of V are orthogonal and V is an n-by-m matrix that satisfies ${V}^{H}V={I}_{m}$.

• For full decompositions, svd(A) returns V as an n-by-n unitary matrix satisfying $V{V}^{H}={V}^{H}V={I}_{n}$. The columns of V that do not correspond to nonzero singular values form a set of orthonormal basis vectors for the null space of A.