# Eigenvalues: Spectral Decomposition

knitr::opts_chunk$set( warning = FALSE, message = FALSE ) options(digits=4) library(matlib) # use the package ## Setup This vignette uses an example of a$3 \times 3$matrix to illustrate some properties of eigenvalues and eigenvectors. We could consider this to be the variance-covariance matrix of three variables, but the main thing is that the matrix is square and symmetric, which guarantees that the eigenvalues,$\lambda_i$are real numbers, and non-negative,$\lambda_i \ge 0$. A <- matrix(c(13, -4, 2, -4, 11, -2, 2, -2, 8), 3, 3, byrow=TRUE) A Get the eigenvalues and eigenvectors using eigen(); this returns a named list, with eigenvalues named values and eigenvectors named vectors. We call these L and V here, but in formulas they correspond to a diagonal matrix,$\mathbf{\Lambda} = diag(\lambda_1, \lambda_2, \lambda_3)$, and a (orthogonal) matrix$\mathbf{V}$. ev <- eigen(A) # extract components (L <- ev$values) (V <- ev$vectors) ## Matrix factorization 1. Factorization of A: A = V diag(L) V'. That is, the matrix$\mathbf{A}$can be represented as the product$\mathbf{A}= \mathbf{V} \mathbf{\Lambda} \mathbf{V}'$. V %*% diag(L) %*% t(V) 1. V diagonalizes A: L = V' A V. That is, the matrix$\mathbf{V}$transforms$\mathbf{A}$into the diagonal matrix$\mathbf{\Lambda}$, corresponding to orthogonal (uncorrelated) variables. diag(L) zapsmall(t(V) %*% A %*% V) ## Spectral decomposition The basic idea here is that each eigenvalue--eigenvector pair generates a rank 1 matrix,$\lambda_i \mathbf{v}_i \mathbf{v}_i '$, and these sum to the original matrix,$\mathbf{A} = \sum_i \lambda_i \mathbf{v}_i \mathbf{v}_i '$. A1 = L * V[,1] %*% t(V[,1]) A1 A2 = L * V[,2] %*% t(V[,2]) A2 A3 = L * V[,3] %*% t(V[,3]) A3 Then, summing them gives A, so they do decompose A: A1 + A2 + A3 all.equal(A, A1+A2+A3) ### Further properties 1. Sum of squares of A = sum of sum of squares of 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)) 2. Each squared eigenvalue gives the sum of squares accounted for by the latent vector L^2 cumsum(L^2) # cumulative 3. The first$i$eigenvalues and vectors give a rank$i\$ approximation to A

R(A1) R(A1 + A2) R(A1 + A2 + A3) # two dimensions sum((A1+A2)^2) sum((A1+A2)^2) / sum(A^2) # proportion