If a matrix A has a non-invertible speed camera matrix P (for example, the matrix [1 1 1; 0 1] has the non-invertible speed camera system [1 0; 0 0]), then A has no speed camera decomposition. However, if A is a real matrix m×n with m>n, then A can be written using a so-called singular decomposition value of the form

A=UDV^(T).

(1)

It should be noted that a number of contrasting notionistic conventions are in use in the literature. Press et al. (1992) define U as m×n matrix, D as n×n and V as n×n. However, the Wolfram language defines U as m×m, D as m×n and V as n×n. In both systems, U and V have orthogonal columns so that

U^(T)U=I

(2)

e

V^(T)V=I

(3)

(where the two identity matrices may have different sizes), and D has entries only along the diagonal.

For a complex matrix A, the decomposition of the singular value is a decomposition into the form

A=UDV^(H),

(4)

where U and V are unit matrixes, V^(H) is the conjugated transposition of V, and D is a diagonal matrix whose elements are the singular values of the original matrix. If A is a complex matrix, then there is always such a decomposition with positive singular values (Golub and Van Loan 1996, pp. 70 and 73).

The decomposition of singular values is implemented in the Wolfram language as SingularValueDecomposition[m], which returns a list {U, D, V}, where U and V are matrices and D is a diagonal matrix composed by the singular values of m.