How to get exact number for repeated eigenvalue in MATLAB and R

Question

Following matrix has repeated eigenvalue $1$.

$$ \begin{pmatrix} 8 & -2 & 1 & -1 \\ 23 & -6 & 3 & -3 \\ -4 & 1 & 1 & 1 \\ -4 & 2 & 0 & 1 \end{pmatrix} $$

How to get $1$ instead of: $$ \underbrace{ \begin{pmatrix} 0.9997 + 0.0000i \\ 1.0000 + 0.0003i \\ 1.0000 - 0.0003i \\ 1.0003 + 0.0000i \end{pmatrix}}_{\text{MATLAB: eig(A)}} \qquad\qquad \underbrace{ \begin{pmatrix} 1.0002024+0.0002024i \\ 1.0002024-0.0002024i \\ 0.9997976+0.0002024i \\ 0.9997976-0.0002024i \end{pmatrix}}_{\text{R: eigen(A)}} $$

Answer 1

You're working with floating point arithmetic, so unless you use a symbolic package, there's going to be some incorrect digits. You could set a tolerance and round to the nearest whole number, but that may impact your computation of other eigenvalues. In Matlab, you could round to the second digit by way of:

round(1.003 + 0.0031i,2)

Edit: why the downvote, was something I said incorrect?

Answer 2

For Octave users, OctSymPy is a symbolic package which try to be compatible with SMT unless otherwise specified. It also claims to work on Matlab, but I don't use Matlab, so I cannot test it. (Please file a bug if it doesn't!)

To run your example, simply type (This should also work on SMT):

eig(sym([8 -2 1 -1; 23 -6 3 -3; -4 1 1 1; -4 2 0 1]))

A screenshot showing the result:

Feel free to ask if you don't understand something.