Eigenvalues and Eigenvectors: Properties (original) (raw)

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 issquare 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
##      [,1] [,2] [,3]
## [1,]   13   -4    2
## [2,]   -4   11   -2
## [3,]    2   -2    8

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)
## [1] 17  8  7
##         [,1]    [,2]   [,3]
## [1,]  0.7454  0.6667 0.0000
## [2,] -0.5963  0.6667 0.4472
## [3,]  0.2981 -0.3333 0.8944

The eigenvalues are always returned in decreasing order, and each column of vectors corresponds to the elements invalues.

Properties of eigenvalues and eigenvectors

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 eigenvalues, \(\lambda_1 \ge \lambda_2 \ge \dots \lambda_p\).

  1. Orthogonality: Eigenvectors are always orthogonal, \(V' V = I\). zapsmall() is handy for cleaning up tiny values.
##           [,1]      [,2]      [,3]
## [1,] 1.000e+00 3.053e-16 5.551e-17
## [2,] 3.053e-16 1.000e+00 0.000e+00
## [3,] 5.551e-17 0.000e+00 1.000e+00
zapsmall(crossprod(vectors))
##      [,1] [,2] [,3]
## [1,]    1    0    0
## [2,]    0    1    0
## [3,]    0    0    1
  1. trace(A) = sum of eigenvalues, \(\sum \lambda_i\).
library(matlib)   # use the matlib package
tr(A)
## [1] 32
## [1] 32
  1. sum of squares of A = sum of squares of eigenvalues, \(\sum \lambda_i^2\).
## [1] 402
## [1] 402
  1. determinant = product of eigenvalues, \(\det(A) = \prod \lambda_i\). This means that the determinant will be zero if any \(\lambda_i = 0\).
## [1] 952
## [1] 952
  1. rank = number of non-zero eigenvalues
## [1] 3
## [1] 3
  1. eigenvalues of the inverse \(A^{-1}\) = 1/eigenvalues of A. The eigenvectors are the same, except for order, because eigenvalues are returned in decreasing order.
##          [,1]    [,2]     [,3]
## [1,]  0.08824 0.02941 -0.01471
## [2,]  0.02941 0.10504  0.01891
## [3,] -0.01471 0.01891  0.13340
## [1] 0.14286 0.12500 0.05882
##        [,1]    [,2]    [,3]
## [1,] 0.0000  0.6667  0.7454
## [2,] 0.4472  0.6667 -0.5963
## [3,] 0.8944 -0.3333  0.2981
  1. There are similar relations for other powers of a matrix, \(A^2, \dots A^p\):values(mpower(A,p)) = values(A)^p, wherempower(A,2) = A %*% A, etc.
## eigen() decomposition
## $values
## [1] 289  64  49
## 
## $vectors
##         [,1]    [,2]   [,3]
## [1,]  0.7454  0.6667 0.0000
## [2,] -0.5963  0.6667 0.4472
## [3,]  0.2981 -0.3333 0.8944
eigen(A %*% A %*% A)$values
## [1] 4913  512  343
eigen(mpower(A, 4))$values
## [1] 83521  4096  2401