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Another use is to find the minimum norm solution to a system of linear equations with multiple solutions. The pseudoinverse facilitates the statement and proof of results in linear algebra. The pseudoinverse is defined for all rectangular matrices whose entries are real or complex numbers. Given a rectangular matrix with real or complex entries ...
where X + is the Moore–Penrose pseudoinverse of X. Although this equation is correct and can work in many applications, it is not computationally efficient to invert the normal-equations matrix (the Gramian matrix). An exception occurs in numerical smoothing and differentiation where an analytical expression is required.
In mathematics, a block matrix pseudoinverse is a formula for the pseudoinverse of a partitioned matrix. This is useful for decomposing or approximating many algorithms updating parameters in signal processing , which are based on the least squares method.
The purpose of constructing a generalized inverse of a matrix is to obtain a matrix that can serve as an inverse in some sense for a wider class of matrices than invertible matrices. Generalized inverses can be defined in any mathematical structure that involves associative multiplication, that is, in a semigroup .
This is applied, e.g., in the Kalman filter and recursive least squares methods, to replace the parametric solution, requiring inversion of a state vector sized matrix, with a condition equations based solution. In case of the Kalman filter this matrix has the dimensions of the vector of observations, i.e., as small as 1 in case only one new ...
In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices. [1] [2]Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or partition it, into a collection of smaller matrices.
is solvable. If the subspace is a proper subspace of , then the matrix of the unconstrained problem () may be singular even if the system matrix of the constrained problem is invertible (in that case, =). This means that one needs to use a generalized inverse for the solution of the constrained problem.
Note that when D is the identity matrix I and +, then = (+) = (/ +) (), therefore the direction of Δ approaches the direction of the negative gradient . The so-called Marquardt parameter λ {\displaystyle \lambda } may also be optimized by a line search, but this is inefficient, as the shift vector must be recalculated every time λ ...