Here is my revision version on the document of the Eigensystem diagonalization example:

In[136]:= (*Let T be the linear transformation whose matrix is given by the matrix a under a intial basis basOrig. 
Find the change-of-basis matrix B for \[DoubleStruckCapitalR]^4 with the property that the representation of T in the final basis basOrth is diagonal:*)
Unprotect[a,b];
a ={{-6, 4, 0, 9},{-3, 0, 1, 6},{-1, -2, 1, 0},{-4, 4, 0, 7}}

(*Find the eigenvalues and eigenvectors of a:   *)
{vals, vecs} = Eigensystem[a]

(*Let B consist of the eigenvectors, and let b be the matrix whose columns are the elements of B:*)

(*b converts the inital basis basOrig to the new orthogonal basis basOrth. Its inverse converts in the reverse direction:*)
b = Transpose[vecs]
bInv = Inverse[b]
basOrth=DiagonalMatrix[vals]//Eigenvectors//Transpose
basOrig=basOrth . bInv
basOrig . b==basOrth

(*Thus the representation of T under the new basis basOrth is given by b^-1.a.b, which is diagonal:*)
bInv . a . b
(*Note that this is simply the diagonal matrix whose entries are the eigenvalues:*)
% == DiagonalMatrix[vals]

Out[137]= {{-6, 4, 0, 9}, {-3, 0, 1, 6}, {-1, -2, 1, 0}, {-4, 4, 0, 7}}

Out[138]= {{5, -2, -2, 
  1}, {{2, 1, -1, 2}, {6, -3, 0, 4}, {1, 1, 1, 0}, {2, -1, -7, 2}}}

Out[139]= {{2, 6, 1, 2}, {1, -3, 1, -1}, {-1, 0, 1, -7}, {2, 4, 0, 2}}

Out[140]= {{-(5/14), 2/7, 1/14, 3/4}, {2/21, -(1/7), 1/21, 0}, {17/21,
   2/7, -(2/21), -1}, {1/6, 0, -(1/6), -(1/4)}}

Out[141]= {{1, 0, 0, 0}, {0, 0, 1, 0}, {0, 1, 0, 0}, {0, 0, 0, 1}}

Out[142]= {{-(5/14), 2/7, 1/14, 3/4}, {17/21, 2/7, -(2/21), -1}, {2/
  21, -(1/7), 1/21, 0}, {1/6, 0, -(1/6), -(1/4)}}

Out[143]= True

Out[144]= {{5, 0, 0, 0}, {0, -2, 0, 0}, {0, 0, -2, 0}, {0, 0, 0, 1}}

Out[145]= True

In short, the role of the eigen basis of matrix a in this example is the change-of-basis matrix.

Regards,
Zhao