## Below is a 3x3 matrix, G.  Let T be the LefthandAction(G).  We'll discover
that T has a single eigenvalue, call it q, of algebraic-multiplicity 3.
However the dimension of the q-eigenspace of T will turn out to be 2-dim'al.
I.e, the geometric-multiplicity of eval q is 2.


> G := Matrix(3,3,{(1, 1) = 3, (1, 2) = 2, (1, 3) = 1, (2, 1) = -80, (2, 2) = -34,
       (2, 3) = -16, (3, 1) = 135, (3, 2) = 54, (3, 3) = 25}) ;

                                [  3      2      1]
                                [                 ]
                           G  = [-80    -34    -16]
                                [                 ]
                                [135     54     25]

> Det(G) ;
                                      -8

> Gofx := G - (x * eye(3)) ;  fGx := Det(Gofx) ; fGx := factor(fGx) ;

                             [3 - x       2         1   ]
                             [                          ]
                     Gofx  = [ -80     -34 - x     -16  ]
                             [                          ]
                             [ 135       54       25 - x]

                                               2    3
                        fGx :=  -8 - 12 x - 6 x  - x

                                          3
                        fGx  =   -(2 + x)

## Ah ha, so the above-mentioned eigenvalue, q, is negative-two.

> M := G - ((-2) * eye(3)) ;   DetM := Det(M) ;

                                [  5      2      1]
                                [                 ]
                           M  = [-80    -32    -16]
                                [                 ]
                                [135     54     27]

                                   DetM = 0

> rrefM := RREF(M) ;
                                   [1    2/5    1/5]
                                   [               ]
                          rrefM  = [0     0      0 ]
                                   [               ]
                                   [0     0      0 ]

> evecn2a := <-2/5, 1, 0>;  evecn2b := <-1/5, 0, 1>;

		  [-2/5]                  [-1/5]
       evecn2a  = [ 1  ] ,     evecn2b  = [ 0  ]
		  [ 0  ]                  [ 1  ]

> evecn2a  := 5 * evecn2a;  evecn2b  := 5 * evecn2b;

		   [-2]                   [-1] 
	evecn2a  = [ 5] ,      evecn2b  = [ 0]
		   [ 0]	                  [ 5]

> GTimesEveca := G . evecn2a; GTimesEvecb := G . evecn2b;

		  [  4]                  [  2]
   GTimesEveca  = [-10] , GTimesEvecb  = [  0]
		  [  0]	                 [-10]

> C := <evecn2a | evecn2b | <0,0,1>>; Dmat := MI(C) . G . C ;

	[-2    -1    0]             [-2     0    -16/5]
	[             ]	            [                 ]
   C  = [ 5     0    0] ,   Dmat  = [ 0    -2    27/5 ]
	[             ]	            [                 ]
	[ 0     5    1]	            [ 0     0     -2  ]


## Perhaps I can make this an all-integer matrix.  Let's change
the third basis-vector.

> C := <evecn2a | evecn2b | <0,0,5>>; Dmat := MI(C) . G . C ;

	[-2    -1    0]            [-2     0    -16]
	[             ]	           [               ]
   C  = [ 5     0    0] ,  Dmat  = [ 0    -2     27]
	[             ]	           [               ]
	[ 0     5    5]	           [ 0     0     -2]

> CDCinv := C . Dmat . MI(C) ;    G := G ;

	  [  3      2      1]         [  3      2      1]
	  [                 ]	      [                 ]
 CDCinv = [-80    -34    -16] ,   G = [-80    -34    -16]
	  [                 ]	      [                 ]
	  [135     54     25]	      [135     54     25]

> Det(C) ;
                                      25

> quit ;
bytes used=4468028, alloc=2948580, time=1.16