vignettes/eigen-ex2.Rmd
eigen-ex2.Rmd
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, \(\lambda_i \ge 0\).
## [,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
. 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$values)
## [1] 17 8 7
(V <- ev$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
## [,1] [,2] [,3]
## [1,] 13 -4 2
## [2,] -4 11 -2
## [3,] 2 -2 8
diag(L)
## [,1] [,2] [,3]
## [1,] 17 0 0
## [2,] 0 8 0
## [3,] 0 0 7
## [,1] [,2] [,3]
## [1,] 17 0 0
## [2,] 0 8 0
## [3,] 0 0 7
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, \(\mathbf{A} = \sum_i \lambda_i \mathbf{v}_i \mathbf{v}_i '\).
## [,1] [,2] [,3]
## [1,] 9.444 -7.556 3.778
## [2,] -7.556 6.044 -3.022
## [3,] 3.778 -3.022 1.511
## [,1] [,2] [,3]
## [1,] 3.556 3.556 -1.7778
## [2,] 3.556 3.556 -1.7778
## [3,] -1.778 -1.778 0.8889
## [,1] [,2] [,3]
## [1,] 0 0.0 0.0
## [2,] 0 1.4 2.8
## [3,] 0 2.8 5.6
Then, summing them gives A
, so they do decompose
A
:
A1 + A2 + A3
## [,1] [,2] [,3]
## [1,] 13 -4 2
## [2,] -4 11 -2
## [3,] 2 -2 8
all.equal(A, A1+A2+A3)
## [1] TRUE
sum(A^2)
## [1] 402
## [1] 289 64 49
## [1] 402
## [1] 402
L^2
## [1] 289 64 49
cumsum(L^2) # cumulative
## [1] 289 353 402
A
R(A1)
## [1] 1
R(A1 + A2)
## [1] 2
R(A1 + A2 + A3)
## [1] 3
# two dimensions
sum((A1+A2)^2)
## [1] 353
## [1] 0.8781