Matrices in R

Singular value decomposition and determinants

The function svd(M) takes an arbitrary matrix argument, M, and calculates the singular value decomposition of M. This consists of a matrix of orthonormal columns U with the same column space as M, a second matrix of orthonormal columns V whose column space is the row space of M and a diagonal matrix of positive entries D such that M = U %*% D %*% t(V). D is actually returned as a vector of the diagonal elements. The result of svd(M) is actually a list of three components named d, u and v, with evident meanings.

If M is in fact square, then, it is not hard to see that

> absdetM <- prod(svd(M)$d)

calculates the absolute value of the determinant of M. If this calculation were needed often with a variety of matrices it could be defined as an R function

> absdet <- function(M) prod(svd(M)$d)

after which we could use absdet() as just another R function. As a further trivial but potentially useful example, you might like to consider writing a function, say tr(), to calculate the trace of a square matrix. [Hint: You will not need to use an explicit loop. Look again at the diag() function.]

R has a builtin function det to calculate a determinant, including the sign, and another, determinant, to give the sign and modulus (optionally on log scale),