7. Eigenvalues and Eigenvectors: Properties

Michael Friendly

2024年11月24日

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 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

(vectors <- 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

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

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, $\lambda_1 \ge \lambda_2 \ge \dots \lambda_p$.

0. Orthogonality: Eigenvectors are always orthogonal, $V' V = I$. zapsmall() is handy for cleaning up tiny values.

crossprod (vectors)

## [,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

sum (values)

## [1] 32

2. sum of squares of A = sum of squares of eigenvalues, $\sum \lambda_i^2$.

sum (A^2)

## [1] 402

sum (values^2)

## [1] 402

3. determinant = product of eigenvalues, $\det(A) = \prod \lambda_i$. This means that the determinant will be zero if any $\lambda_i = 0$.

det (A)

## [1] 952

prod (values)

## [1] 952

4. rank = number of non-zero eigenvalues

R (A)

## [1] 3

sum (values != 0)

## [1] 3

5. 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.

AI <- solve (A)
AI

## [,1] [,2] [,3]
## [1,] 0.08824 0.02941 -0.01471
## [2,] 0.02941 0.10504 0.01891
## [3,] -0.01471 0.01891 0.13340

eigen (AI)$values

## [1] 0.14286 0.12500 0.05882

eigen (AI)$vectors

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

6. There are similar relations for other powers of a matrix, $A^2, \dots A^p$: values(mpower(A,p)) = values(A)^p, where mpower(A,2) = A %*% A, etc.

eigen (A %*%  A)

## 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