7.3. Eigen Value Decomposition

Eigen value decomposition:

> A <- matrix(1:9, nrow=3)
> A <- A + t(A)
> eigen(A)
eigen() decomposition
$values
[1]  3.291647e+01 -5.329071e-15 -2.916473e+00

$vectors
           [,1]       [,2]       [,3]
[1,] -0.3516251  0.4082483  0.8424328
[2,] -0.5533562 -0.8164966  0.1647127
[3,] -0.7550872  0.4082483 -0.5130074

Let us verify the decomposition:

> A
     [,1] [,2] [,3]
[1,]    2    6   10
[2,]    6   10   14
[3,]   10   14   18


> e <- eigen(A)
> lambda <- diag(e$values)
> U <- e$vectors
> U %*% lambda %*% t(U)
     [,1] [,2] [,3]
[1,]    2    6   10
[2,]    6   10   14
[3,]   10   14   18

Computing only the eigen values:

> eigen(A, only.values = TRUE)
$values
[1]  3.291647e+01 -1.787388e-15 -2.916473e+00

$vectors
NULL