rjd {JADE}R Documentation

Joint Diagonalization of Real Matrices

Description

This is an R version of Cardoso's rjd matlab function for joint diagonalization of k real-valued square matrices.

Usage

rjd(X, eps = 1e-06, maxiter = 100, na.action = na.fail)

Arguments

X A matrix of k stacked pxp matrices with dimension c(kp,p) or an array with dimension c(p,p,k).
eps Convergence tolerance.
maxiter Maximum number of iterations.
na.action A function which indicates what should happen when the data contain 'NA's. Default is to fail.

Details

Denote the square matrices as A_i, i=1,...,k. This algorithm searches then an orthogonal matrix V so that D_i=V'A_iV is diagonal for all i. If the A_i commute then there is an exact solution. If not, the function will perform an approximate joint diagonalization by trying to make the D_i as diagonal as possible.

Cardoso points out that notion of approximate joint diagonalization is ad hoc and very small values of eps make in that case not much sense since the diagonality criterion is ad hoc itself.

Value

A list with the components

V An orthogonal matrix.
D A stacked matrix with the diagonal matrices or an array with the diagonal matrices. The form of the output depends on the form of the input.

Author(s)

Jean-Francois Cardoso. Ported to R by Klaus Nordhausen, klaus.nordhausen@uta.fi

References

Cardoso, J.-F. and Souloumiac, A., (1996), Jacobi angles for simultaneous diagonalization, SIAM J. Mat. Anal. Appl., 17, 161–164.

Examples

Z <- matrix(runif(9), ncol = 3)
U <- eigen(Z %*% t(Z))$vectors
D1 <- diag(runif(3))
D2 <- diag(runif(3))
D3 <- diag(runif(3))
D4 <- diag(runif(3))

X.matrix <- rbind(t(U) %*% D1 %*% U, t(U) %*% D2 %*% U,
                  t(U) %*% D3 %*% U, t(U) %*% D4 %*% U)
res.matrix <- rjd(X.matrix)
res.matrix$V
round(U %*% res.matrix$V, 4) # should be a signed permutation matrix if V is correct.

round(res.matrix$D, 4)

X.array <- aperm(array(t(X.matrix), dim = c(3,3,4)), c(2,1,3))

res.array <- rjd(X.array)
round(res.array$D, 4)

[Package JADE version 1.0-1 Index]