The singular value decomposition of an real matrix
(1)
In the full decomposition, orthogonal matrix ,
orthogonal matrix , and matrix with nonnegative singular
values of
There are several notational conventions in use. A reduced decomposition uses diagonal matrix ,
and orthonormal , so that
(2)
and
(3)
for the appropriate identity matrices (Golub and
Van Loan 1996, pp. 70-71).
For a complex matrix
(4)
where unitary matrices , conjugate transpose
of nonnegative singular
values in positions zero elsewhere. Such a decomposition
exists for every real matrix or complex
matrix , including rectangular matrices .
The full decomposition gives eigenvalues of this Gram
matrix are the squared singular values of real matrix ,
the conjugate transpose is replaced by the
transpose (Watkins 2005).
When coefficient matrix of a vector
in a tensor product of two Hilbert
spaces , its singular value decomposition gives the Schmidt
decomposition of that vector.
Singular value decomposition is implemented in the Wolfram Language as SingularValueDecomposition m ],
which returns a list {U , D , V }. Here
U , D , and V are matrices , and the
only possibly nonzero entries of D are the singular values of
See also Cholesky Decomposition ,
Eigen Decomposition ,
Eigen
Decomposition Theorem ,
Eigenvalue ,
Eigenvector ,
Gram Matrix ,
LU Decomposition ,
Matrix Decomposition ,
QR
Decomposition ,
Schmidt Decomposition ,
Singular Value ,
Unitary
Matrix
Explore with Wolfram|Alpha
References Drmač, Z. and Veselić, K. "New Fast and Accurate Jacobi SVD Algorithm. I." SIAM J. Matrix Anal. Appl. 29 , 1322-1342,
2008a. https://doi.org/10.1137/050639193 . Drmač,
Z. and Veselić, K. "New Fast and Accurate Jacobi SVD Algorithm. II."
SIAM J. Matrix Anal. Appl. 29 , 1343-1362, 2008b. https://doi.org/10.1137/05063920X . Gentle,
J. E. "Singular Value Factorization." §3.2.7 in Numerical
Linear Algebra for Applications in Statistics. Golub, G. and Kahan, W. "Calculating the
Singular Values and Pseudo-Inverse of a Matrix." J. Soc. Indust. Appl. Math.
Ser. B: Numer. Anal. 2 , 205-224, 1965. https://doi.org/10.1137/0702016 . Golub,
G. H. and Reinsch, C. "Singular Value Decomposition and Least Squares Solutions."
Numer. Math. 14 , 403-420, 1970. https://doi.org/10.1007/BF02163027 . Golub,
G. H. and Van Loan, C. F. "The Singular Value Decomposition"
and "Unitary Matrices." §2.5.3 and 2.5.6 in Matrix
Computations, 3rd ed. Nash, J. C. "The Singular-Value Decomposition
and Its Use to Solve Least-Squares Problems." Ch. 3 in Compact
Numerical Methods for Computers: Linear Algebra and Function Minimisation, 2nd ed. Press, W. H.;
Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Singular
Value Decomposition." §2.6 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Stewart, G. W.
"On the Early History of the Singular Value Decomposition." SIAM Rev. 35 ,
551-566, 1993. https://doi.org/10.1137/1035134 . Watkins,
D. S. "Product Eigenvalue Problems." SIAM Rev. 47 , 3-40,
2005. https://doi.org/10.1137/S0036144504443110 . Referenced
on Wolfram|Alpha Singular Value Decomposition
Cite this as:
Weisstein, Eric W. "Singular Value Decomposition."
From MathWorld https://mathworld.wolfram.com/SingularValueDecomposition.html
Subject classifications
real matrix 
is a factorization
is an 
orthogonal matrix,
is an 
orthogonal matrix, and 
is an 
matrix with 
for 
.
Its entries 
, 
,
are the nonnegative singular
values of 
.
and takes 
to be 
, 
to be a 
diagonal matrix,
and 
to be 
. In either convention, the columns retained
in 
and 
are orthonormal, so that
, the full singular value decomposition has the form
and 
are unitary matrices, 
is the conjugate transpose
of 
, and 
again has nonnegative singular
values in positions 
and zero elsewhere. Such a decomposition
exists for every real matrix or complex
matrix, including rectangular matrices.
.
Thus the eigenvalues of this Gram
matrix are the squared singular values of 
, together with additional zeros when 
. For a real matrix,
the conjugate transpose is replaced by the
transpose (Watkins 2005).
is the coefficient matrix of a vector
in a tensor product of two Hilbert
spaces, its singular value decomposition gives the Schmidt
decomposition of that vector.
in positions 
.
