Synonyms for pseudoinverse or Related words with pseudoinverse
Examples of "pseudoinverse"
In mathematics, and in particular linear algebra, a
of a matrix is a generalization of the inverse matrix. The most widely known type of matrix
is the Moore–Penrose
, which was independently described by E. H. Moore in 1920, Arne Bjerhammar in 1951 and Roger Penrose in 1955. Earlier, Erik Ivar Fredholm had introduced the concept of a
of integral operators in 1903. When referring to a matrix, the term
, without further specification, is often used to indicate the Moore–Penrose
. The term generalized inverse is sometimes used as a synonym for
The singular value decomposition can be used for computing the
of a matrix. Indeed, the
of the matrix with singular value decomposition is
"A" is called the Moore-Penrose
of "A". Notice that A is also the Moore-Penrose
of "A" . That is, ("A" )"" = A.
Now solving for formula_27 yields the
Another method for computing the
uses the recursion
We now show that formula_6 is a
Note that formula_9 is the
We now show that formula_16 is a
where is the
of , which is formed by replacing every non-zero diagonal entry by its reciprocal and transposing the resulting matrix. The
is one way to solve linear least squares problems.
of the null (all zero) vector is the transposed null vector. The
of a non-null vector is the conjugate transposed vector divided by its squared magnitude:
To avoid problems when and/or do not exist, division can also be defined as multiplication with the
, i.e., and , where and denote the
of "A" and "B".
In mathematics, block matrix
is a formula of
of a partitioned matrix. This is useful for decomposing or approximating many algorithms updating parameters in signal processing, which are based on least squares method.
This can be proven by defining matrices formula_88, formula_89, and checking that formula_90 is indeed a
for formula_91 by verifying that the defining properties of the
hold, when formula_13 is Hermitian and idempotent.
The MASS package for R provides a calculation of the Moore–Penrose
through the codice_5 function. The codice_5 function calculates a
using the singular value decomposition provided by the codice_7 function in the base R package.
It is also possible to define a
for scalars and vectors. This amounts to treating these as matrices. The
of a scalar is zero if is zero and the reciprocal of otherwise:
where formula_6 is the Moore-Penrose
of "A" and "w" is any "n"×1 vector.
provides a least squares solution to a system of linear equations.
The ordinary least squares solution is to estimate the coefficient vector using the Moore-Penrose
For the case where is not invertible the Moore–Penrose
should be used instead.
constitutes a "left inverse", since, in this case, formula_34.
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