Why my matrix inverse implementation is less accurate? C++

Question

I have a transformation matrix of type Eigen::Matrix4d. And I would like to get its inverse. I write a function my self to compute it by the following formular.

.

And here is my code:

Eigen::Matrix4d inverseTransformation(Eigen::Matrix4d T)
{
    Eigen::MatrixX3d R;
    Eigen::Vector3d t;
    R = T.block<3, 3>(0, 0);
    t = T.block<3, 1>(0, 3);
    Eigen::Matrix4d result;
    result.setIdentity();
    result.block<3, 3>(0, 0) = R.transpose();
    result.block<3, 1>(0, 3) = -R.transpose() * t;
    return result;
}



    std::cout<<"Input transformation matrix is:\n" << T << std::endl;
    std::cout << "inverse of T , my implementation:\n" << inverseTransformation(T) << std::endl << std::endl;
    std::cout << "inverse of T , Eigen implementation::\n" << T.inverse() << std::endl << std::endl;
    std::cout << "T * T^(-1), my implementation\n " << T * inverseTransformation(T) << std::endl << std::endl;
    std::cout << "T * T^(-1), eigen's implementation\n " << T * T.inverse() << std::endl << std::endl;

Ideally, T * T^(1) should be I. However, there is some error in my result (the red part in the following picture.)

By contrast, the result from T * T.inverse() is much more accurate.

Could someone please tell me why? Thanks a lot in advance!

Update: Inverse of a rotation matrix is its transpose. The result will be more accurate if I replace R.tranpose() with R.inverse().

Answer 1

Thanks to the comments. Since I would like to close this question, I will answer it my self.

Theoretically, R.inverse() equals to R.tranpose(). But this is not correct in floating point system. Once I use R.inverse(), the result is more accurate.