svd

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

[U,S,V] = svd(A) performs a singular value decomposition of matrix A, such that A = U*S*V'. S is a diagonal matrix of the same dimension as A with nonnegative diagonal elements in decreasing order. U and V are unitary matrices.

[___] = svd(A,"econ") produces an economy-size decomposition of A. If A is an m-by-n matrix, then:

m > n — Only the first n columns of U are computed and S is n-by-n.
m = n — svd(A,"econ") is equivalent to svd(A).
m < n — Only the first m columns of V are computed, and S is m-by-m.

[___] = svd(A,0) produces a different economy-size decomposition of A. If A is an m-by-n matrix, then:

m > n — svd(A,0) is equivalent to svd(A,"econ").
m <= n — svd(A,0) is equivalent to svd(A).

Syntax is not recommended. Use the "econ" option instead.

[___] = svd(___,sigmaForm) optionally specifies the output format for the singular values. You can use this option with any of the previous input or output combinations. Specify "vector" to return the singular values as a column vector. Specify "matrix" to return the singular values in a diagonal matrix.

Examples

Singular Values of Fixed-Point Matrix

Compute the singular values of a full rank fixed-point matrix.

A = fi([1 0 1; -1 -2 0; 0 1 -1])

A = 
     1     0     1
    -1    -2     0
     0     1    -1

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 16
        FractionLength: 14

Compute the singular values.

s = svd(A)

s = 
    2.4605
    1.6996
    0.2392

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 16

The singular values are returned in a column vector in decreasing order.

Fixed-Point Singular Value Decomposition

Find the singular value decomposition of the rectangular fixed-point matrix A.

Define the rectangular matrix A.

m = 4;
n = 2;
rng('default');
A = fi(10*randn(m,n))

A = 
    5.3770    3.1875
   18.3389  -13.0771
  -22.5889   -4.3359
    8.6221    3.4258

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 16
        FractionLength: 10

Find the singular value decomposition of the fixed-point matrix A.

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

U = 
    0.1591    0.2717   -0.9387   -0.1403
    0.6397   -0.7548   -0.1219    0.0790
   -0.7049   -0.5057   -0.3224    0.3786
    0.2619    0.3174         0    0.9114

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 30

S = 
   31.0148         0
         0   14.1290
         0         0
         0         0

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 16

V = 
    0.9920    0.1259
   -0.1259    0.9920

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 30

Confirm the relation A = U*S*V'.

U*S*V'

ans = 
    5.3770    3.1873
   18.3390  -13.0773
  -22.5890   -4.3360
    8.6221    3.4257

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 99
        FractionLength: 76

Economy-Size Decomposition

Calculate the complete and economy-size decomposition of a rectangular fixed-point matrix.

Define the fixed-point matrix A.

m = 5;
n = 3;
rng('default');
A = fi(10*randn(m,n))

A = 
    5.3770  -13.0762  -13.4980
   18.3379   -4.3359   30.3496
  -22.5879    3.4258    7.2539
    8.6211   35.7832   -0.6309
    3.1875   27.6953    7.1465

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 16
        FractionLength: 9

Compute the complete decomposition.

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

U = 
    0.3081   -0.0950    0.4507    0.7929    0.2534
   -0.1437    0.9533   -0.0877    0.2415   -0.0675
   -0.0224   -0.2106   -0.8423    0.4887   -0.0831
   -0.7299   -0.1909    0.2773    0.2722   -0.5290
   -0.5926   -0.0375   -0.0541         0    0.8028

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 30

S = 
   48.4483         0         0
         0   36.6720         0
         0         0   26.9112
         0         0         0
         0         0         0

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 16

V = 
   -0.1786    0.5444    0.8196
   -0.9497   -0.3131    0.0009
   -0.2571    0.7783   -0.5729

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 30

Compute the economy-size decomposition.

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

U = 
    0.3081   -0.0950    0.4507
   -0.1437    0.9533   -0.0878
   -0.0224   -0.2106   -0.8423
   -0.7299   -0.1909    0.2773
   -0.5926   -0.0374   -0.0541

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 30

S = 
   48.4485         0         0
         0   36.6720         0
         0         0   26.9112

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 16

V = 
   -0.1786    0.5444    0.8196
   -0.9497   -0.3131    0.0010
   -0.2571    0.7783   -0.5729

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 30

Use the expected result A = U*S*V' to determine the relative error of the calculation.

relativeError = norm(double(U*S*V'-A))/norm(double(A))

relativeError = 1.0359e-05

Control Singular Value Output Format

Create a 3-by-3 magic square matrix and calculate the singular value decomposition. By default, the svd function returns the singular values in a diagonal matrix when you specify multiple outputs.

Define the matrix A.

m = 3; 
n = m;
A = fi(magic(m))

A = 
     8     1     6
     3     5     7
     4     9     2

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 16
        FractionLength: 11

Compute the singular value decomposition.

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

U = 
    0.5774   -0.7071   -0.4083
    0.5773   -0.0000    0.8165
    0.5773    0.7071   -0.4082

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 30

S = 
   15.0000         0         0
         0    6.9283         0
         0         0    3.4642

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 16

V = 
    0.5774   -0.4082   -0.7071
    0.5773    0.8165    0.0000
    0.5773   -0.4082    0.7071

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 30

Specify the "vector" option to return the singular values in a column vector.

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

U = 
    0.5774   -0.7071   -0.4083
    0.5773   -0.0000    0.8165
    0.5773    0.7071   -0.4082

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 30

S = 
   15.0000
    6.9283
    3.4642

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 16

V = 
    0.5774   -0.4082   -0.7071
    0.5773    0.8165    0.0000
    0.5773   -0.4082    0.7071

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 32
        FractionLength: 30

If you specify one output argument, such as S = svd(A), then svd switches behavior to return the singular values in a column vector by default. In that case, you can specify the "matrix" option to return the singular values as a diagonal matrix.

Input Arguments

`A` — Input matrix
matrix

Input matrix, specified as a matrix. A can be a fixed-point or scaled double fi data type.

Data Types: fi
Complex Number Support: Yes

`sigmaForm` — Output format of singular values
`"vector"` | `"matrix"`

Output format of singular values, specified as one of these values:

"vector" — S is a column vector. This behavior is the default when you specify one output, S = svd(A).
"matrix" — S is a diagonal matrix. This behavior is the default when you specify multiple outputs, [U,S,V] = svd(A).

Example: [U,S,V] = svd(X,"vector") returns S as a column vector instead of a diagonal matrix.

Example: S = svd(X,"matrix") returns S as a diagonal matrix instead of a column vector.

Data Types: char | string

Output Arguments

`U` — Left singular vectors
matrix

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

The fixed-point data type is adjusted to avoid overflow and increase precision. For more information, see Algorithms.

`S` — Singular values
diagonal matrix | column vector

Singular values, returned as a diagonal matrix or column vector. The singular values are nonnegative and returned in decreasing order.

The fixed-point data type is adjusted to avoid overflow and increase precision. For more information, see Algorithms.

`V` — Right singular vectors
matrix

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

The fixed-point data type is adjusted to avoid overflow and increase precision. For more information, see Algorithms.

Tips

To have full control over the fixed-point types, use the fixed.svd function.

Algorithms