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The Octave programming language provides a pseudoinverse through the standard package function pinv and the pseudo_inverse() method. In Julia (programming language), the LinearAlgebra package of the standard library provides an implementation of the Moore–Penrose inverse pinv() implemented via singular-value decomposition. [25]
Instead of ([] []), we need to calculate directly or indirectly [citation needed] [original research? (), (), (), ().In a dense and small system, we can use singular value decomposition, QR decomposition, or Cholesky decomposition to replace the matrix inversions with numerical routines.
In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element x is an element y that has some properties of an inverse element but not necessarily all of them. 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 ...
When the channel state information is perfectly known at the transmitter, the zero-forcing precoder is given by the pseudo-inverse of the channel matrix. Zero-forcing has been used in LTE mobile networks. [2]
With the inverse of A available, it is only necessary to find the inverse of C −1 + VA −1 U in order to obtain the result using the right-hand side of the identity. If C has a much smaller dimension than A, this is more efficient than inverting A + UCV directly.
Pseudo-inverse The pseudo inverse of a matrix is the unique matrix = + for which and are symmetric and for which =, = holds. If is nonsingular, then + =. When ...
Inverse transformation of Vaníček's LSSA is possible, as is most easily seen by writing the forward transform as a matrix; the matrix inverse (when the matrix is not singular) or pseudo-inverse will then be an inverse transformation; the inverse will exactly match the original data if the chosen sinusoids are mutually independent at the ...
This is the inverse function theorem. Furthermore, if the Jacobian determinant at p is positive , then f preserves orientation near p ; if it is negative , f reverses orientation. The absolute value of the Jacobian determinant at p gives us the factor by which the function f expands or shrinks volumes near p ; this is why it occurs in the ...