Essential matrix is not new to anybody being familiar with Epipolar Geometry. Here is the famous epipolar constraint:
\[ x_2^\top E x_1 = 0 \]Recall that when the essential matrix is obtained by 8-point algorithm, one has to decompose the result with SVD to make its third singular value 0 and the other two singular values equal. Here is a short derivation of this property of essential matrix.
According to the definition of essential matrix:
\[ E = \{ S R | S = t^\wedge \in \mathfrak{so}(3), R \in SO(3)\} \tag{1-1} \]According to the definition of SVD, a singular value \(\sigma\) of a matrix A, equals to the square root of correspondent eigenvalue \(\lambda\) of matrix \(A^\top A\):
\[ \sigma_{A,i} = \sqrt{\lambda_{A^\top A,i}} \tag{1-2} \]Meanwhile,
\[ E^\top E = R^\top S^\top S R = -R^\top (t^\wedge)^2 R \tag{1-3} \]According to the property of askew-symmetric matrix:
\[ (t^\wedge)^2 = \|t\|^2 I - t t^\top \tag{1-4} \]The first term is a scalar matrix with three equal eigenvalues \(\|t\|^2\)
The second term \(t t^\top\) is a symmetric matrix, which is therefore semi-positive definite. It can be diagonalized[1]:
\[ t t^\top = \|t\|^2 R_t^\top diag(1, 0, 0) R_t \tag{1-5} \]Thus, equation (1-3) becomes:
\[ E^\top E = \|t\|^2 R^\top R_t^\top ( I - diag(1, 0, 0)) R_t R \] \[ = \|t\|^2 (R_t R )^\top diag(1, 1, 0) (R_t R) \tag{1-6} \]Therefore, eigenvalues of \(E^\top E\) are: \(\|t\|^2 (1, 1, 0)\).
Finally, according to (1-3), singular values of essential matrix \(E\) are: \(\|t\|(1, 1, 0)\).