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Generalized singular value decomposition
[U,V,X,C,S] = gsvd(A,B) [U,V,X,C,S] = gsvd(A,B,0) sigma = gsvd(A,B)
[U,V,X,C,S] = gsvd(A,B)
returns unitary matrices U and V, a (usually) square matrix X, and nonnegative diagonal matrices C and S so that
A = U*C*X' B = V*S*X' C'*C + S'*S = I
A and B must have the same number of columns, but may have different numbers of rows. If A is m-by-p and B is n-by-p, then U is m-by-m, V is n-by-n and X is p-by-q where q = min(m+n,p).
sigma = gsvd(A,B)
returns the vector of generalized singular values, sqrt(diag(C'*C)./diag(S'*S)).
The nonzero elements of S are always on its main diagonal. If m >= p the nonzero elements of C are also on its main diagonal. But if m < p, the nonzero diagonal of C is diag(C,p-m). This allows the diagonal elements to be ordered so that the generalized singular values are nondecreasing.
gsvd(A,B,0), with three input arguments and either m or n >= p, produces the "economy-sized" decomposition where the resulting U and V have at most p columns, and C and S have at most p rows. The generalized singular values are diag(C)./diag(S).
When B is square and nonsingular, the generalized singular values, gsvd(A,B), are equal to the ordinary singular values, svd(A/B), but they are sorted in the opposite order. Their reciprocals are gsvd(B,A).
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 are equal to svd([A;B]) and cond([A;B]). Other formulations, eg. G. Golub and C. Van Loan [1], require that null(A) and null(B) do not overlap and replace X by 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.
In the first example, the matrices have at least as many rows as columns.
A = reshape(1:15,5,3)
B = magic(3)
A =
1 6 11
2 7 12
3 8 13
4 9 14
5 10 15
B =
8 1 6
3 5 7
4 9 2
The statement
[U,V,X,C,S] = gsvd(A,B)produces a 5-by-5 orthogonal
U, a 3-by-3 orthogonal V, a 3-by-3 nonsingular X,
X =
-2.8284 9.3761 -6.9346
5.6569 8.3071 -18.3301
-2.8284 7.2381 -29.7256
and
C =
0.0000 0 0
0 0.3155 0
0 0 0.9807
0 0 0
0 0 0
S =
1.0000 0 0
0 0.9489 0
0 0 0.1957
Since A is rank deficient, the first diagonal element of C is zero.
The economy sized decomposition,
[U,V,X,C,S] = gsvd(A,B,0)produces a 5-by-3 matrix
U and a 3-by-3 matrix C.
U =
-0.3736 -0.6457 -0.4279
-0.0076 -0.3296 -0.4375
0.8617 -0.0135 -0.4470
-0.2063 0.3026 -0.4566
-0.2743 0.6187 -0.4661
C =
0.0000 0 0
0 0.3155 0
0 0 0.9807
The other three matrices, V, X, and S are the same as those obtained with the full decomposition.
The generalized singular values are the ratios of the diagonal elements of C and S.
sigma = gsvd(A,B)
sigma =
0.0000
0.3325
5.0123
These values are a reordering of the ordinary singular values
svd(A/B)
ans =
5.0123
0.3325
0.0000
In the second example, the matrices have at least as many columns as rows.
A = reshape(1:15,3,5)
B = magic(5)
A =
1 4 7 10 13
2 5 8 11 14
3 6 9 12 15
B =
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
The statement
[U,V,X,C,S] = gsvd(A,B)produces a 3-by-3 orthogonal
U, a 5-by-5 orthogonal V, a 5-by-5 nonsingular X and
C =
0 0 0.0000 0 0
0 0 0 0.0439 0
0 0 0 0 0.7432
S =
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
In this situation, the nonzero diagonal of C is diag(C,2). The generalized singular values include three zeros.
sigma = gsvd(A,B)
sigma =
0
0
0.0000
0.0439
1.1109
Reversing the roles of A and B reciprocates these values, producing three infinities.
gsvd(B,A)
ans =
0.9001
22.7610
Inf
Inf
Inf
The generalized singular value decomposition uses the C-S decomposition described in [1], as well as the built-in svd and qr functions. The C-S decomposition is implemented in a subfunction in the gsvd M-file.
The only warning or error message produced by gsvd itself occurs when the two input arguments do not have the same number of columns.
[1] Golub, Gene H. and Charles Van Loan, Matrix Computations, Third Edition, Johns Hopkins University Press, Baltimore, 1996
svd Singular value decomposition