Example 9.   Find the eigenvalues and eigenvectors of the matrix  .

Solution 9.

Mathematica did not make a mistake !  We just don't know what it meant !  Just ask it for those numbers !  

We could have avoided the problem completely by just using the numerical features of Mathematica.  

Symbolic solution of the eigenvectors for this example can be done, however the numerical solution is easier to read.

Here is the documentation for Mathematica's subroutine Solve.  
You can find it by going to the Help menu and locating help for GroebnerBasis,
then scroll down and go to: Implementation Notes: see section A.9.5.  

Exact equation solving
1.    For linear equations Gaussian elimination and other methods of linear algebra are used.
2.    Root objects representing algebraic numbers are usually isolated and manipulated using validated numerical methods.
    With  ExactRootIsolation->True,  Root  uses for real roots a continued fraction version of an algorithm based on Descartes' rule of signs,
    and for complex roots the Collins­Krandick algorithm.
3.    For single polynomial equations, Solve uses explicit formulas up to degree four,attempts to reduce polynomials using Factor and Decompose,
    and recognizes cyclotomic and other special polynomials.
4.    For systems of polynomial equations, Solve constructs a Gröbner basis.
5.    Solve and GrobnerBasis use an efficient version of the Buchberger algorithm.
6.    For non­polynomial equations, Solve attempts to change variables and add polynomial side conditions.
7.    The code inside Mathematica for Solve is about 500 pages long.

Comments.  How much do we know or teach about polynomials ?  How much should we ?  The area of computer science called artificial intelligence treats the topic of "expert system."  To construct an "expert system" you tap into the brains of the experts, and program them into your computer.  Apparently Mathematica has already done this, it took 500 lines of code.  They speak of "information overload" in the future.  The future was 16 years ago with Mathematica ©1988 and 23 years ago with Maple ©1981.

(c) John H. Mathews 2004