JADE (version 1.1-0)

rjd: 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.

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.

Details

Denote the square matrices as $A_i$, $i=1,\ldots,k$. 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.

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)