knitr::opts_chunk$set( warning = FALSE, message = FALSE ) options(digits=4)
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. Covariance matrices are also positive semi-definite, meaning that their eigenvalues are 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
.
ev <- eigen(A) # extract components (values <- ev$values) (vectors <- ev$vectors)
The eigenvalues are always returned in decreasing order, and each column of vectors
corresponds to the
elements in values
.
The following steps illustrate the main properties of eigenvalues and eigenvectors. We use the notation $A = V' \Lambda V$ to express the decomposition of the matrix $A$, where $V$ is the matrix of eigenvectors and $\Lambda = diag(\lambda_1, \lambda_2, \dots, \lambda_p)$ is the diagonal matrix composed of the ordered eivenvalues, $\lambda_1 \ge \lambda_2 \ge \dots \lambda_p$.
zapsmall()
is handy for cleaning up tiny values.crossprod(vectors) zapsmall(crossprod(vectors))
library(matlib) # use the matlib package tr(A) sum(values)
sum(A^2) sum(values^2)
det(A) prod(values)
R(A) sum(values != 0)
AI <- solve(A) AI eigen(AI)$values eigen(AI)$vectors
values(mpower(A,p)) = values(A)^p
, where
mpower(A,2) = A %*% A
, etc.eigen(A %*% A) eigen(A %*% A %*% A)$values eigen(mpower(A, 4))$values
Any scripts or data that you put into this service are public.
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.