sirt (version 1.5-0)

eigenvalues.manymatrices: Computation of Eigenvalues of Many Symmetric Matrices

Description

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 <- eigenvalues.manymatrices( Sigma.all )
m1

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