How to obtain right eigenvectors of matrix in R?

Question

Edition : the problem in my question was I've tried to find matrix `S` from equation 8 but this equation have error. How to directly obtain right eigenvectors of matrix in R ? 'eigen()' gives only left eigenvectors Really last edition, I've made big mess here, but this question is really important for me : `eigen()` provides some matrix of eigenvectors, from function help : " If ‘r <- eigen(A)’, and ‘V <- r$vectors; lam <- r$values’, then ``` A = V Lmbd V^(-1) ``` (up to numerical fuzz), where `Lmbd =diag(lam)`" that is `A V = V Lmbd`, where V is matrix now we check it : ``` set.seed(1) A<-matrix(rnorm(16),4,4) Lmbd=diag(eigen(A)$values) V=eigen(A)$vectors A%*%V > A%*%V [,1] [,2] [,3] [,4] [1,] 0.0479968+0.5065111i 0.0479968-0.5065111i 0.2000725+0i 0.30290103+0i [2,] -0.2150354+1.1746298i -0.2150354-1.1746298i -0.4751152+0i -0.76691563+0i [3,] -0.2536875-0.2877404i -0.2536875+0.2877404i 1.3564475+0i 0.27756026+0i [4,] 0.9537141-0.0371259i 0.9537141+0.0371259i 0.3245555+0i -0.03050335+…

Accepted Answer

A worked example.

Default (= right eigenvectors):

```
m <- matrix(1:9,nrow=3)
e <- eigen(m)
e1 <- e$vectors
zapsmall((m %*% e1)/e1) ## right e'vec
##          [,1]      [,2] [,3]
## [1,] 16.11684 -1.116844    0
## [2,] 16.11684 -1.116844    0
## [3,] 16.11684 -1.116844    0
```

Left eigenvectors:

```
eL <- eigen(t(m))    
eL1 <- eL$vectors
```

(We have to go to a little more effort since we need to be multiplying by row vectors on the left; if we extracted just a single eigenvector, R's ignorance of row/column vector distinctions would make it "do the right thing" (i.e. `(eL1[,1] %*% m)/eL1[,1]` just works).)

```
zapsmall(t(eL1) %*% m/(t(eL1)))
##          [,1]      [,2]      [,3]
## [1,] 16.116844 16.116844 16.116844
## [2,] -1.116844 -1.116844 -1.116844
## [3,]  0.000000  0.000000  0.000000
```

How to obtain right eigenvectors of matrix in R?

Solution

Problem