;; Given a list G1, G2, G3, ...   of vectors, we compute
; an orthogonal system
;              B1, B2, B3, ...  with the same span.


;; A more interesting example, in 4-dim'al space.


% (setq G1 (mat-make-initseq 4 1 '(1 0 3 0)))
    [  1  ]
    [  0  ]
    [  3  ]
    [  0  ]

% (setq G2 (mat-make-initseq 4 1 '(4 2 2 -2)))
    [   4  ]
    [   2  ]
    [   2  ]
    [  -2  ]

% (setq G3 (mat-make-initseq 4 1 '(-1 0 -1 2)))
    [  -1  ]
    [   0  ]
    [  -1  ]
    [   2  ]


;; Technically, B1 is the orthogonal-vector of G1 
;; with respect to the trivial-subspace.
% (setq B1 G1)
    [  1  ]
    [  0  ]
    [  3  ]
    [  0  ]

% (setq B2 (mat-sub G2 (proj G2 B1))) ;Is Orth_B1(G2)
    [   3  ]
    [   2  ]
    [  -1  ]
    [  -2  ]

;; This next is the orthogonal-vector of G3
;; w.r.t Span(B1, B2).

% (setq B3
     (mat-sub G3  (mat-add (proj G3 B1) (proj G3 B2))) )
    [   2/5  ]
    [   2/3  ]
    [  -2/15 ]
    [   4/3  ]


;; Let's multiply to make all the entries integers.
% (setq B3 (mat-scal-mult 15 B3))
    [   6  ]
    [  10  ]
    [  -2  ]
    [  20  ]


;; We verify that the pairwise inner-products of the 
;; B-vectors are zero:

% (list (ip B1 B2) (ip B1 B3) (ip B2 B3)) -> (0 0 0)

;; Let's check that Span(B1, B2, B3) = Span(G1, G2, G3).
% (setq bob (mat-Horiz-concat B1 B2 B3 G1 G2 G3))
    [  1   3   6  1   4  -1  ]
    [  0   2  10  0   2   0  ]
    [  3  -1  -2  3   2  -1  ]
    [  0  -2  20  0  -2   2  ]

% (rref-mtab-beforecol bob)
  JK: Found 3 pivots before the sixth column.

       c0   c1   c2   c3   c4   c5
   |---------------------------------|
r0 |    1    0    0    1    1  -2/5  |
r1 |    0    1    0    0    1  -1/3  |
r2 |    0    0    1    0    0   1/15 |
r3 |    0    0    0    0    0    0   | ; Yay, Gram-Schmidt.
   |---------------------------------|