orblsam2-Theoretical basis (III.)

See ORBSLAM2 initialization in the INITIALIZER::RECONSTRUCTH and INITIALIZER::RECONSTRUCTF two sub-functions used in the OPENCV::SVD decomposition. Here I will explain in detail the decomposition theory of SVD!

singular value decomposition (Singular value decomposition) is an important matrix decomposition in linear algebra

Suppose M is a MXN-order matrix in which all elements belong to the domain K, that is, the real field or the plural field. So there is a decomposition that makesM = uσv*,where U is a MXM-order unitary matrix; σ is a semi-definite MXN-order diagonal matrix, and v*, or v conjugate transpose, is the NXN-order unitary matrix. Such decomposition is called the singular value decomposition of M. The element σi,i on the σ diagonal is the singular value of M. It is common practice for singular values to be arranged in large and small order. So σ can be determined by the M single. (Although U and V are still not deterministic.) )

Where the unitary matrix is defined as:

The N-column vector of n-order Compound array U is a standard orthogonal base of u space, then u is the unitary matrix (unitary matrix). Obviously the unitary matrix is the generalization of the orthogonal matrix in the reciprocating number field.

The element on the diagonal of the matrix σ equals the singular value of M. The columns of U and v are singular vectors of the left and right, respectively, of singular values. Therefore, the above theorem shows that:a matrix of MXN has at most a different singular value of p = min (m,n). It is always possible to find an orthogonal base u in km, which consists of the left singular vector of M. It is always possible to find an orthogonal base V with a kn that makes up the right singular vector of M.
U is the M x m matrix, where U is listed as mm ^{The orthogonal eigenvector of T, V is n x n matrix, where V is listed as Mtm orthogonal eigenvectors, assuming that R is the rank of M-matrix, there is singular value decomposition:
M = uσv* (v* is conjugate transpose of V)
where mmt and mtm has the same singular value (if it is a real number, it has the same eigenvalue)}

In the homogeneous equation

The method of least squares for other non-homogeneous equations

orblsam2-Theoretical basis (III.)