[?] Get eigenvalues of a real symmetric matrix?

Questions
,
1

Hello,

I am using Wolfram Alpha widgets for calculating Eigen values of a 3X3 real symmetric matrix. It is known the a real symmetric matrix has real Eigen values.
However, the calculator is giving roots with iota (complex looking roots). So I have two questions,

  1. Are the Eigen values computed by the widget actually complex?

  2. If eigen values are not complex, then why aren’t they simplified and given? How do I simplify them?

The matrix which I used is :

A = [  [ 1    2    3],
  [2   1   2],
  [3   2   3] ]

Here is the link also,

Thanks in advance. Hoping to hear from you soon.

Sandhya

P.S. : For the above matrix A, real Eigen values are 6.569, -1.342, -0.227.

2

It looks like the results are correct. For example, finding the numerical value of the first eigenvalue given shows it is real.

In[5]:= 1/
  3 (5 + 55/(377 + 3 I Sqrt[2694])^(1/3) + (377 + 3 I Sqrt[2694])^(
    1/3)) // N

Out[5]= 6.56872 - 3.33067*10^-16 I
3

Mathematica tries to preserve infinite precision when it can. Use floating point numbers in your matrix like 1.0, 2.0 etc., and Eigenvalues will return numbers instead of expressions.