STA: "IP-PosDefinite.2x2.txt"		13Nov2023:
  Given colvecs b0 and b1 (defined below),
we compute the unique
positive-definite matrix P such that 
the inner-product  (acting on colvecs)

    (c0, c1) |->   TPose(c0) * P * c1

makes {b0,b1} an ortho-normal basis of |R^2.


                LISP CODE
;; For testing purposes, define the std basis:
(setq   e0 (mat-make-colvec 1 0)   e1 (mat-make-colvec 0 1))
    [ 1 ]      [ 0 ]
    [ 0 ] ,    [ 1 ]

(setq b0 (mat-make-colvec 1 3)     b1 (mat-make-colvec 1 -1))
    [ 1 ]      [  1 ]
    [ 3 ] ,    [ -1 ]  


;; This will realize B-orthonormal IP, once I compute matrix P.
(defun IP-by-P (c0 c1) (mat-E (mat-mul (mat-tpose c0) P c1) 0 0))


;; Computing 2x2 matrix P:
(setq Id-ToE-FrB (mat-Horiz-concat b0 b1)) =>  [ 1   1 ]
                                               [ 3  -1 ]
    
(setq Id-ToB-FrE (mat-inverse? Id-ToE-FrB))
    1      [ 1   1 ]                                     
   ---  *  [       ]
    4      [ 3  -1 ]

;; Let's check that we have the correct CoB matrix.
(mat-mul Id-ToB-FrE  b0)   =>  [ 1 ]
                               [ 0 ]
    
(mat-mul Id-ToB-FrE  b1)   =>  [ 0 ]
                               [ 1 ]
    

(setq P (mat-mul (mat-tpose Id-ToB-FrE) Id-ToB-FrE))
            [  5  -1 ]       [  5/8  -1/8 ]
    (1/8) * [        ]   =   [            ]
            [ -1   1 ]       [ -1/8   1/8 ]

(IP-by-P b0 b1)  =>  0

(IP-by-P b0 b0)  =>  1

(IP-by-P b1 b1)  =>  1

;; Nifty.

END: "IP-PosDefinite.2x2.txt"