Determining if a matrix is diagonalizable in the R Programming Language

Question

I have a matrix and I would like to know if it is diagonalizable. How do I do this in the R programming language?

Answer 1

If you have a given matrix, m, then one way is the take the eigen vectors times the diagonal of the eigen values times the inverse of the original matrix. That should give us back the original matrix. In R that looks like:

m <- matrix( c(1:16), nrow = 4)
p <- eigen(m)$vectors
d <- diag(eigen(m)$values)
p %*% d %*% solve(p)
m

so in that example p %*% d %*% solve(p) should be the same as m

Answer 2

You can implement the full algorithm to check if the matrix reduces to a Jordan form or a diagonal one (see e.g., this document). Or you can take the quick and dirty way: for an n-dimensional square matrix, use eigen(M)$values and check that they are n distinct values. For random matrices, this always suffices: degeneracy has prob.0.

P.S.: based on a simple observation by JD Long below, I recalled that a necessary and sufficient condition for diagonalizability is that the eigenvectors span the original space. To check this, just see that eigenvector matrix has full rank (no zero eigenvalue). So here is the code:

diagflag = function(m,tol=1e-10){
    x = eigen(m)$vectors
    y = min(abs(eigen(x)$values))
    return(y>tol)
}
# nondiagonalizable matrix 
m1 = matrix(c(1,1,0,1),nrow=2) 
# diagonalizable matrix
m2 = matrix(c(-1,1,0,1),nrow=2) 

> m1
     [,1] [,2]
[1,]    1    0
[2,]    1    1

> diagflag(m1)
[1] FALSE

> m2
     [,1] [,2]
[1,]   -1    0
[2,]    1    1

> diagflag(m2)
[1] TRUE