How to prove the existence of solution of a non linear system of equationsWriting the ortogonality condition for any element of O ( n ), I've arrived to:If we take n = 2, we know that Λ Λ T = I, so: ( x y z t ) ( x z y t ) = ( 1 0 0 1 ) This yields: { x 2 + y 2 = 1 x z + y t = 0 z 2 + t 2 = 1 Visually, these equations mean that we can find two orthonormal vectors.We could generalise our reasoning to arbitrary dimension easily.How could I prove rigorously, without plugging in any numbers nor functions, that this system of equations has solution (infinite, in fact)?Is there any method to prove the existence of solution of non non linear equations (or systems)?

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How to prove the existence of solution of a non linear system of equationsWriting the ortogonality condition for any element of   O  (  n  ), I&#039;ve arrived to:If we take   n  =  2, we know that   Λ      Λ      T    =  I, so:      (                            x                          y                                      z                          t                      )        (                            x                          z                                      y                          t                      )    =      (                            1                          0                                      0                          1                      )  This yields:      {                                        x                      2                    +                      y                      2                    =          1                                      x          z          +          y          t          =          0                                                  z                      2                    +                      t                      2                    =          1                        Visually, these equations mean that we can find two orthonormal vectors.We could generalise our reasoning to arbitrary dimension easily.How could I prove rigorously, without plugging in any numbers nor functions, that this system of equations has solution (infinite, in fact)?Is there any method to prove the existence of solution of non non linear equations (or systems)?

Mollie Roberts · Accepted Answer

The first and third equations states that points   (  x  ,  y  ) and   (  z  ,  t  ) are on a unit circle. Let&#039;s write that as:   x  =  cos  ⁡  ϕ  y  =  sin  ⁡  ϕ  z  =  cos  ⁡  ψ  t  =  sin  ⁡  ψNow we may write the second equation as:  cos  ⁡  ϕ  cos  ⁡  ψ  +  sin  ⁡  ϕ  sin  ⁡  ψ  =  0This may be rewritten as:  cos  ⁡  (  ϕ  −  ψ  )  =  0Which has a solution:  ϕ  −  ψ  =      π    2    +  k  πWe should restrict the angles to   (  −  π  ,  π  ):  ϕ  =  ψ  ±      π    2  Which means that for each    (  x  ,  y  ) on a unit circle there are two possible   (  z  ,  t  ).

Jaeden Weaver · Accepted Answer

Your system for arbitrary dimension n is equivalent to finding an orthonormal basis in         R    n   (put its elements as solumns in your matrix), which, in its turn is trivial.

How to prove the existence of solution of a non linear system of equations Writing the ortogonality

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