# Function for Joint Diagonalization of k Square Matrices in a Deflation Based Manner This function jointly diagonalizes k real-valued square matrices by searching an orthogonal matrix in a deflation based manner. ```r djd(X, G = "max", r = 2, eps = 1e-06, maxiter = 500) ``` ## 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 `max`, `pow` and `log`. See details. - `r`: power value used if `G="pow"` or `G="max"`. 0 is not meaningful for this value. See details. - `eps`: convergence tolerance. - `maxiter`: maximum number of iterations. ## Details Denote the square matrices as $A_i$, $i=1,...,k$. 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 function G can be choosen to be of the form $G(x) = |x|^r$ or $G(x) = log(x)$. If `G="max"` is chosen, the function G is of the form $G(x) = |x|^r$, and the diagonalization criterion will be maximized globally at each stage by choosing an appropriate initial value from a set of random vectors. If `G="pow"` or `G="log"` are chosen, the initial values are the eigenvectors of $A_1$ which plays hence a special role. ## Returns The matrix W ## References Nordhausen, K., Gutch, H. W., Oja, H. and Theis, F.J. (2012): JointDiagonalization of Several Scatter Matrices for ICA, in _LVA/ICA 2012_,LNCS 7191, pp. 172-179. Miettinen, J., Nordhausen, K., Oja, H. and Taskinen, S. (2014),Deflation-based Separation of Uncorrelated Stationary Time Series,_Journal of Multivariate Analysis_, 123, 214-227. Miettinen, J., Nordhausen, K. and Taskinen, S. (2017), Blind SourceSeparation Based on Joint Diagonalization in R: The Packages JADE andBSSasymp, _Journal of Statistical Software_, *76*, 1-31,\. ## Author(s) Klaus Nordhausen, Jari Miettinen ## Examples ```r 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. ```

djd function