The hardware and bandwidth for this mirror is donated by METANET, the Webhosting and Full Service-Cloud Provider.
If you wish to report a bug, or if you are interested in having us mirror your free-software or open-source project, please feel free to contact us at mirror[@]metanet.ch.
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, \(\lambda_i \ge 0\).
## [,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
.
## [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 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.## [,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
## [,1] [,2] [,3]
## [1,] 1 0 0
## [2,] 0 1 0
## [3,] 0 0 1
## [1] 32
## [1] 32
## [1] 402
## [1] 402
## [1] 952
## [1] 952
## [1] 3
## [1] 3
## [,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
values(mpower(A,p)) = values(A)^p
, where
mpower(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
## [1] 4913 512 343
## [1] 83521 4096 2401
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.