Eigenvalues: Spectral Decomposition"

  warning = FALSE,
  message = FALSE
library(matlib)   # use the package


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)

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.
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])

A2 = L[2] * V[,2] %*% t(V[,2])

A3 = L[3] * V[,3] %*% t(V[,3])

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
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)
  1. Each squared eigenvalue gives the sum of squares accounted for by the latent vector
cumsum(L^2)   # cumulative
  1. The first $i$ eigenvalues and vectors give a rank $i$ approximation to A
R(A1 + A2)
R(A1 + A2 + A3)
# two dimensions
sum((A1+A2)^2) / sum(A^2)   # proportion

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matlib documentation built on April 4, 2018, 5:03 p.m.