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[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

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