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The orthogonal Procrustes problem [1] is a matrix approximation problem in linear algebra.In its classical form, one is given two matrices and and asked to find an orthogonal matrix which most closely maps to .
The matrix X is subjected to an orthogonal decomposition, e.g., the QR decomposition as follows. = , where Q is an m×m orthogonal matrix (Q T Q=I) and R is an n×n upper triangular matrix with >. The residual vector is left-multiplied by Q T.
Mathematically, linear least squares is the problem of approximately solving an overdetermined system of linear equations A x = b, where b is not an element of the column space of the matrix A. The approximate solution is realized as an exact solution to A x = b', where b' is the projection of b onto the column space of A. The best ...
In mathematics, low-rank approximation refers to the process of approximating a given matrix by a matrix of lower rank. More precisely, it is a minimization problem, in which the cost function measures the fit between a given matrix (the data) and an approximating matrix (the optimization variable), subject to a constraint that the approximating matrix has reduced rank.
This algorithm takes a finite number of steps to reach a solution and smoothly improves its candidate solution as it goes (so it can find good approximate solutions when cut off at a reasonable number of iterations), but is very slow in practice, owing largely to the computation of the pseudoinverse ((A P) T A P) −1. [1]
Now () is an matrix solution of ′ =. This fundamental matrix will provide the homogeneous solution, and if added to a particular solution will give the general solution to the inhomogeneous equation. Let = be the general solution. Now,
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The matrix multiplication () yields the required square matrix and the matrix-vector product on the right hand side yields a vector of size . The result is a set of n {\displaystyle n} linear equations, which can be solved for δ {\displaystyle {\boldsymbol {\delta }}} .