JADE (version 1.1-0)

djd: Function for Joint Diagonalization of k Square Matrices in a Deflation Based Manner

Description

This function jointly diagonalizes k real-valued square matrices by searching an orthogonal matrix in a deflation based manner.

Usage

djd(X, G = "pow", r = 2, eps = 1e-06, maxiter = 100)

Arguments

X

an array containing the k p times p real valued matrices of dimension c(p, p, k).

G

criterion function used for the the algorithm. Options are pow and log. See details.

r

power value used if G="pow". 0 is not meaningful for this value. See details.

eps

convergence tolerance.

maxiter

maximum number of iterations.

Value

• The matrix W

Details

Denote the square matrices as $A_i$, $i=1,\ldots,k$. This algorithm searches then an orthogonal matrix W so that $D_i=W'A_iW$ 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 maximizing $\sum G(w_j' A_i w_j)$ where $w_j$ are the orthogonal vectors in W. The directions are found in a deflation based manner where the initial values are the eigenvectors of $A_1$ which plays hence a special role. The function G can be choosen to be of the form $G(x) = x^r$ or $G(x) = log(x)$.

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 <- array(0, dim=c(3, 3, 4))
X.matrix[,,1] <- t(U) %*% D1 %*% U
X.matrix[,,2] <- t(U) %*% D2 %*% U
X.matrix[,,3] <- t(U) %*% D3 %*% U
X.matrix[,,4] <- t(U) %*% D4 %*% U

W1 <- djd(X.matrix)
round(U %*% W1, 4) # should be a signed permutation 
                     # matrix if W1 is correct.

W2 <- djd(X.matrix, r=1)
round(U %*% W2, 4) # should be a signed permutation 
                     # matrix if W2 is correct.

W3 <- djd(X.matrix, G="l")
round(U %*% W3, 4) # should be a signed permutation 
                     # matrix if W3 is correct.