Jacobi Plane rotation for a matrix A

How to perform the Jacobi Rotation (Jacobi Method) for given matrix

There exists a rotation (c = cos(theta) and s = sin(theta))

For example how can we estimate rotation for the following matrix

A = [-17.7147 -38.4117  30.6475
     -51.3024  17.3859 -10.0354
     -19.3323 -38.8931  30.3686
     -51.2891  18.9043 -11.1523
     -21.42 -39.2796  29.9065
     -51.1701  20.7146 -12.4891
     -24.2543 -39.5276  29.3515
     -51.0782  22.9095 -14.1458]

Using C++ Eigen libaray the result is following: http://eigen.tuxfamily.org/dox-3.2/classEigen_1_1JacobiRotation.html

 Result = [ 110.564    -7.77137   -0.308057
            0      87.445    -64.7691
            0           0     1.86159
            0           0           0
            0           0           0
            0           0           0
            0           0           0
            0           0           0]

Using matlab inbuilt function qr ([~,R]=qr(A)) gives me the following:

R = [  110.5645   -7.7714   -0.3081
         0  -87.4451         64.7691
         0         0        -1.8616
         0         0         0
         0         0         0
         0         0         0
         0         0         0
         0         0         0]        

As you can see the first row result is same in C++ and matlab.

whereas the second and third row signs are not matching.

What is the correct solution or how can it be implemented in matlab?

Thank you!!

Answer 1

[KSSV]

Define the ngle of your rotation theta:

A = [-17.7147 -38.4117  30.6475
     -51.3024  17.3859 -10.0354
     -19.3323 -38.8931  30.3686
     -51.2891  18.9043 -11.1523
     -21.42 -39.2796  29.9065
     -51.1701  20.7146 -12.4891
     -24.2543 -39.5276  29.3515
     -51.0782  22.9095 -14.1458] ; 
theta = pi/4 ;
R = [cos(theta) sin(theta) 0 ; 
     -sin(theta) cos(theta) 0 ; 
    0   0   1] ; 
Ar = A*R

[Ravindra]

So for Jacobi Method we start from the bottom of the matrix with values -24.2543 and -51.0782. if we correctly estimate the cos and sin angles the result should be

[c,s] = -0.4289    0.9033

The rotation matrix will be

[ -0.4289   -0.9033
  0.9033   -0.4289]

The A matrix is modified using this rotation and will be:

  -17.7147  -38.4117   30.6475
  -51.3024   17.3859  -10.0354
  -19.3323  -38.8931   30.3686
  -51.2891   18.9043  -11.1523
  -21.4200  -39.2796   29.9065
  -51.1701   20.7146  -12.4891
   56.5443   -3.7398    0.1882
         0  -45.5334   32.5819

The next step will be with values -51.1701 and 56.5443

[c,s] = 0.6710    0.7415        

in C++ at this step the estimated c & s both have negative signs

-0.6710    -0.7415 % in C++ 

The problem is the Jacobi rotation matrices from C++ are different what I calculate with my own algorithm.

This results in different answers. I am not sure which is correct.