In friendly/matlib: Matrix Functions for Teaching and Learning Linear Algebra and Multivariate Statistics

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 guarrantees 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))

1. Each squared eigenvalue gives the sum of squares accounted for by the latent vector

L^2
cumsum(L^2)   # cumulative

1. 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

friendly/matlib documentation built on March 3, 2024, 12:18 p.m.