(original) (raw)

## ----echo = FALSE------------------------------------------------------------- knitr::opts_chunk$set( warning = FALSE, message = FALSE ) options(digits=4) ## ----------------------------------------------------------------------------- library(matlib) # use the package ## ----------------------------------------------------------------------------- A <- matrix(c(13, -4, 2, -4, 11, -2, 2, -2, 8), 3, 3, byrow=TRUE) A ## ----------------------------------------------------------------------------- ev <- eigen(A) # extract components (L <- ev$values) (V <- ev$vectors) ## ----------------------------------------------------------------------------- V %*% diag(L) %*% t(V) ## ----------------------------------------------------------------------------- diag(L) zapsmall(t(V) %*% A %*% V) ## ----------------------------------------------------------------------------- A1 = L[1] * V[,1] %*% t(V[,1]) A1 A2 = L[2] * V[,2] %*% t(V[,2]) A2 A3 = L[3] * V[,3] %*% t(V[,3]) A3 ## ----------------------------------------------------------------------------- A1 + A2 + A3 all.equal(A, A1+A2+A3) ## ----------------------------------------------------------------------------- sum(A^2) c( sum(A1^2), sum(A2^2), sum(A3^2) ) sum( sum(A1^2), sum(A2^2), sum(A3^2) ) #' same as tr(A' A) tr(crossprod(A)) ## ----------------------------------------------------------------------------- L^2 cumsum(L^2) # cumulative ## ----------------------------------------------------------------------------- R(A1) R(A1 + A2) R(A1 + A2 + A3) # two dimensions sum((A1+A2)^2) sum((A1+A2)^2) / sum(A^2) # proportion