Mathc matrices/c29a4

La décomposition spectral

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	c04.c

/* ------------------------------------ */
/*  Save as :  c04.c                    */
/* ------------------------------------ */
#include "v_a.h"
/* ------------------------------------ */
void fun(int r)
{
double **A         = rsymmetric_mR(i_mR(r,r),999.);
double **EigsVector=               i_mR(r,r);
double **EigsVector_T=             i_mR(r,r);

double **b1=                       i_mR(r,C1);
double **b2=                       i_mR(r,C1);
double **b3=                       i_mR(r,C1);

double **r1=                       i_mR(R1,r);
double **r2=                       i_mR(R1,r);
double **r3=                       i_mR(R1,r);

double **b1r1=                     i_mR(r,r);
double **b2r2=                     i_mR(r,r);
double **b3r3=                     i_mR(r,r);

double **T  =                      i_mR(r,r);

  clrscrn();
  printf(" A:");
  p_mR(A, S5, P0, C6);

  eigs_V_mR(A, EigsVector); 
  printf(" EigsVector:");
  p_mR(EigsVector, S5, P6, C6);

  transpose_mR(EigsVector,EigsVector_T);
  printf(" Inverse of EigsVector = EigsVector_T");
  pE_mR(EigsVector_T, S12, P4, C6);
  stop();


  clrscrn(); 
  c_c_mR(EigsVector, C1, b1, C1 );
  c_c_mR(EigsVector, C2, b2, C1 );
  c_c_mR(EigsVector, C3, b3, C1 );
  
  c_r_mR(EigsVector_T, R1, r1, C1 );
  c_r_mR(EigsVector_T, R2, r2, C1 );  
  c_r_mR(EigsVector_T, R3, r3, C1 ); 
  
  mul_mR(b1, r1, b1r1);
  mul_mR(b2, r2, b2r2);
  mul_mR(b3, r3, b3r3); 
  
  mul_mR(b1r1, b2r2, T);  
  printf(" b1r1 * b2r2");
  p_mR(T, S12, P3, C6);
  
  mul_mR(b1r1, b3r3, T);  
  printf(" b1r1 * b3r3");
  p_mR(T, S12, P3, C6);
  
  mul_mR(b2r2, b3r3, T);  
  printf(" b2r2 * b3r3");
  p_mR(T, S12, P3, C6);
  
  f_mR(A);
  f_mR(EigsVector);
  f_mR(EigsVector_T);
    
  f_mR(b1);
  f_mR(b2);
  f_mR(b3);

  f_mR(r1);
  f_mR(r2);
  f_mR(r3);  
 
  f_mR(b1r1);
  f_mR(b2r2);
  f_mR(b3r3);

  f_mR(T);
}
/* ------------------------------------ */
int main(void)
{

do
{
 fun(R3);

} while(stop_w());

  return 0;
}
/* ------------------------------------ */
/* ------------------------------------ */

Nous voyons une des propriétés de la décomposition spectral:

• b1r1 * b2r2 = 0
• b1r1 * b3r3 = 0
• b2r2 * b3r3 = 0

Cela matérialise simplement que nos projecteurs sont orthogonaux entre eux.

Exemple de sortie écran :

 ------------------------------------ 

 A:
 -479  -269  +992 
 -269  -501  +904 
 +992  +904  -763 

 EigsVector:
-0.512981 +0.544352 -0.663726 
-0.485715 +0.453457 +0.747300 
+0.707765 +0.705732 +0.031785 

 Inverse of EigsVector = EigsVector_T
 -5.1298e-01  -4.8571e-01  +7.0777e-01 
 +5.4435e-01  +4.5346e-01  +7.0573e-01 
 -6.6373e-01  +7.4730e-01  +3.1785e-02 

 Press return to continue. 



 b1r1 * b2r2
      +0.000       +0.000       +0.000 
      +0.000       +0.000       +0.000 
      -0.000       -0.000       -0.000 

 b1r1 * b3r3
      +0.000       +0.000       -0.000 
      +0.000       -0.000       -0.000 
      +0.000       +0.000       +0.000 

 b2r2 * b3r3
      +0.000       -0.000       -0.000 
      +0.000       -0.000       -0.000 
      +0.000       -0.000       -0.000 


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