gsvd

Generalized singular value decomposition

Syntax

[U,V,X,C,S] = gsvd(A,B)

[U,V,X,C,S] = gsvd(A,B,"econ")

[U,V,X,C,S] = gsvd(A,B,0)

sigma = gsvd(A,B)

Description

[U,V,X,C,S] = gsvd(A,B) performs a generalized singular value decomposition of matrices A and B, and returns unitary matrices U and V, a matrix X, and nonnegative diagonal matrices C and S such that

A = U*C*X'
B = V*S*X'
C'*C + S'*S = I 

[U,V,X,C,S] = gsvd(A,B,"econ"), where A is m-by-p and B is n-by-p, produces an economy-size decomposition where the resulting U and V matrices have at most p columns, and C and S have at most p rows.

[U,V,X,C,S] = gsvd(A,B,0) is the same as gsvd(A,B,"econ").

sigma = gsvd(A,B) returns the vector of generalized singular values, sqrt(diag(C'*C)./diag(S'*S)). When B is square and nonsingular, the generalized singular values, gsvd(A,B), correspond to the ordinary singular values, svd(A/B), but they are sorted in the opposite order. Their reciprocals are gsvd(B,A).

Examples

Generalized Singular Value Decomposition of Matrices

Perform a generalized singular value decomposition on two matrices and also calculate the generalized singular values.

Create a 5-by-3 matrix A and a 3-by-3 matrix B.

A = reshape(1:15,5,3)

A = 5×3

     1     6    11
     2     7    12
     3     8    13
     4     9    14
     5    10    15

B = magic(3)

B = 3×3

     8     1     6
     3     5     7
     4     9     2

Perform a generalized singular value decomposition on A and B. The outputs include orthogonal U and V, a nonsingular X, and diagonal matrices C and S. Because A is rank deficient, the first diagonal element of C is zero.

[U,V,X,C,S] = gsvd(A,B)

U = 5×5

    0.5618    0.6457   -0.4279   -0.0735    0.2810
   -0.8005    0.3296   -0.4375    0.1240    0.2094
   -0.0219    0.0135   -0.4470   -0.4343   -0.7816
    0.1979   -0.3026   -0.4566    0.7906   -0.1889
    0.0626   -0.6187   -0.4661   -0.4068    0.4802

V = 3×3

   -0.7071   -0.6946    0.1325
    0.0000   -0.1874   -0.9823
    0.7071   -0.6946    0.1325

X = 3×3

   -2.8284   -9.3761   -6.9346
    5.6569   -8.3071  -18.3301
   -2.8284   -7.2381  -29.7256

C = 5×3

    0.0000         0         0
         0    0.3155         0
         0         0    0.9807
         0         0         0
         0         0         0

S = 3×3

    1.0000         0         0
         0    0.9489         0
         0         0    0.1957

Now, perform an economy-size decomposition of A and B. The matrices U and C now have three rows instead of five, but the other output matrices are the same.

[U,V,X,C,S] = gsvd(A,B,"econ")

U = 5×3

   -0.4780    0.6457   -0.4279
    0.7808    0.3296   -0.4375
   -0.3081    0.0135   -0.4470
    0.1857   -0.3026   -0.4566
   -0.1804   -0.6187   -0.4661

V = 3×3

    0.7071   -0.6946    0.1325
    0.0000   -0.1874   -0.9823
   -0.7071   -0.6946    0.1325

X = 3×3

    2.8284   -9.3761   -6.9346
   -5.6569   -8.3071  -18.3301
    2.8284   -7.2381  -29.7256

C = 3×3

    0.0000         0         0
         0    0.3155         0
         0         0    0.9807

S = 3×3

    1.0000         0         0
         0    0.9489         0
         0         0    0.1957

Calculate the generalized singular values, which in this case are equal to the ratios diag(C)./diag(S). These values are a reordering of the ordinary singular values returned by svd(A/B).

sigma = gsvd(A,B)

sigma = 3×1

    0.0000
    0.3325
    5.0123

svals = svd(A/B)

svals = 3×1

    5.0123
    0.3325
    0.0000

Matrices with Infinite Singular Values

Examine why generalized singular values can sometimes be Inf.

Create a 3-by-5 matrix A and a 5-by-5 matrix B.

A = reshape(1:15,3,5)

A = 3×5

     1     4     7    10    13
     2     5     8    11    14
     3     6     9    12    15

B = magic(5)

B = 5×5

    17    24     1     8    15
    23     5     7    14    16
     4     6    13    20    22
    10    12    19    21     3
    11    18    25     2     9

Perform a generalized singular value decomposition on A and B. The outputs include orthogonal U and V, a nonsingular X, and diagonal matrices C and S. In this situation, the nonzero diagonal of C is diag(C,2).

[U,V,X,C,S] = gsvd(A,B)

U = 3×3

    0.4082    0.7178   -0.5639
   -0.8165    0.0109   -0.5772
    0.4082   -0.6961   -0.5906

V = 5×5

   -0.2725   -0.3116   -0.7838   -0.4552   -0.0843
   -0.6610    0.0670    0.5070   -0.5408    0.0956
   -0.0078   -0.0833    0.1573   -0.0271   -0.9836
    0.6218   -0.4832    0.2797   -0.5408    0.0956
    0.3195    0.8111   -0.1602   -0.4552   -0.0843

X = 5×5

  -10.1340         0    0.0000  -30.7287   -4.6958
    3.3206    1.1584  -14.8596  -28.5003  -11.6858
   14.8016   10.1717    6.1195  -26.2720  -18.6758
    2.1067  -11.7464    9.5268  -24.0436  -25.6657
  -10.0949    0.4163   -0.7867  -21.8152  -32.6557

C = 3×5

         0         0    0.0000         0         0
         0         0         0    0.0439         0
         0         0         0         0    0.7432

S = 5×5

    1.0000         0         0         0         0
         0    1.0000         0         0         0
         0         0    1.0000         0         0
         0         0         0    0.9990         0
         0         0         0         0    0.6690

Find the generalized singular values, which include zeros.

sigma = gsvd(A,B)

sigma = 5×1

         0
         0
    0.0000
    0.0439
    1.1109

Reversing the roles of A and B inverts these values, producing Inf values.

sigma2 = gsvd(B,A)

sigma2 = 5×1
1016 ×

    0.0000
    0.0000
    3.2092
       Inf
       Inf

Input Arguments

A, B — Input matrices (as separate arguments)

matrices

Input matrices. A and B must have the same number of columns but can have different numbers of rows.

Data Types: single | double
Complex Number Support: Yes

Output Arguments

U, V — Unitary matrix factors

matrices

Unitary matrix factors, returned as matrices. U and V are matrices with orthogonal columns that provide a basis for the columns of A and B, respectively.

The sizes of U and V depend on whether you specify the "econ" option. If A is m-by-p and B is n-by-p, then:

• By default, U is m-by-m and V is n-by-n.

• If you specify the "econ" option, then U is m-by-min(m,p) and V is n-by-min(n,p).

X — Matrix factor

matrix

Matrix factor, returned as a matrix. If A is m-by-p and B is n-by-p, then X is p-by-q, where q = min(m+n,p).

A property of X is that norm([A; B]) == norm(X) because U, V, C, and S represent only an orthogonal basis for the matrix [A; B].

C, S — Diagonal matrix factors

matrices

Diagonal matrix factors, returned as matrices. The nonzero elements of S are always on its main diagonal. The nonzero elements of C are on the diagonal diag(C,max(0,size(C,2)-size(C,1))). To retrieve the nonzero values, use the commands sv = max(S,[],1) and cv = max(C,[],1). The generalized singular values of A and B are equal to the ratio cv./sv.

The sizes of C and S depend on whether you specify the "econ" option. If A is m-by-p and B is n-by-p, then:

• By default, C is m-by-q and S is n-by-q.

• If you specify the "econ" option, then C is min(m,p)-by-q and V is min(n,p)-by-q.

In both cases, q = min(m+n,p).

sigma — Generalized singular values

column vector

Generalized singular values, returned as a column vector. If sv = max(S,[],1) and cv = max(C,[],1), then the generalized singular values of A and B are equal to the ratio cv./sv.

sigma has length min(m+n,p) and is in nondecreasing order.

Tips

• In this formulation of the gsvd, no assumptions are made about the individual ranks of A or B. The matrix X has full rank if and only if the matrix [A; B] has full rank. In fact, svd(X) and cond(X) are equal to svd([A; B]) and cond([A; B]). Other formulations, such as in G. Golub and C. Van Loan [1], require that null(A) and null(B) do not overlap and replace X with inv(X) or inv(X').

  Note, however, that when null(A) and null(B) do overlap, the nonzero elements of C and S are not uniquely determined.

Algorithms

The generalized singular value decomposition performed by the gsvd function uses the C-S decomposition described in [1], as well as the built-in svd and qr functions.

References

[1] Golub, Gene H., and Charles F. Van Loan. Matrix Computations. 3rd ed. Johns Hopkins Studies in the Mathematical Sciences. Baltimore: Johns Hopkins University Press, 1996.

Extended Capabilities

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

Version History

Introduced before R2006a

R2022a: Option for economy-size decompositions

Use the "econ" option to calculate economy-size decompositions with gsvd. The functionality is the same as gsvd(A,B,0).