Computation of Eigenvalues of Many Symmetric Matrices — eigenvalues.manymatrices • sirt

This function computes the eigenvalue decomposition of $N$ symmetric positive definite matrices. The eigenvalues are computed by the Rayleigh quotient method (Lange, 2010, p. 120). In addition, the inverse matrix can be calculated.

Usage

eigenvalues.manymatrices(Sigma.all, itermax=10, maxconv=0.001,
    inverse=FALSE )

Arguments

Sigma.all: An $N \times D^2$ matrix containing the $D^2$ entries of $N$ symmetric matrices of dimension $D \times D$
itermax: Maximum number of iterations
maxconv: Convergence criterion for convergence of eigenvectors
inverse: A logical which indicates if the inverse matrix shall be calculated

Value

A list with following entries

lambda: Matrix with eigenvalues
U: An $N \times D^2$ Matrix of orthonormal eigenvectors
logdet: Vector of logarithm of determinants
det: Vector of determinants
Sigma.inv: Inverse matrix if inverse=TRUE.

References

Lange, K. (2010). Numerical Analysis for Statisticians. New York: Springer.

Examples

# define matrices
Sigma <- diag(1,3)
Sigma[ lower.tri(Sigma) ] <- Sigma[ upper.tri(Sigma) ] <- c(.4,.6,.8 )
Sigma1 <- Sigma

Sigma <- diag(1,3)
Sigma[ lower.tri(Sigma) ] <- Sigma[ upper.tri(Sigma) ] <- c(.2,.1,.99 )
Sigma2 <- Sigma

# collect matrices in a "super-matrix"
Sigma.all <- rbind( matrix( Sigma1, nrow=1, byrow=TRUE),
                matrix( Sigma2, nrow=1, byrow=TRUE) )
Sigma.all <- Sigma.all[ c(1,1,2,2,1 ), ]

# eigenvalue decomposition
m1 <- sirt::eigenvalues.manymatrices( Sigma.all )
m1

# eigenvalue decomposition for Sigma1
s1 <- svd(Sigma1)
s1