Examples of "pseudoinverse"
In mathematics, and in particular linear algebra, a pseudoinverse of a matrix is a generalization of the inverse matrix. The most widely known type of matrix pseudoinverse is the Moore–Penrose pseudoinverse, 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 pseudoinverse of integral operators in 1903. When referring to a matrix, the term pseudoinverse, without further specification, is often used to indicate the Moore–Penrose pseudoinverse. The term generalized inverse is sometimes used as a synonym for pseudoinverse.
The singular value decomposition can be used for computing the pseudoinverse of a matrix. Indeed, the pseudoinverse of the matrix with singular value decomposition is
"A" is called the Moore-Penrose pseudoinverse of "A". Notice that A is also the Moore-Penrose pseudoinverse of "A" . That is, ("A" )"" = A.
Now solving for formula_27 yields the pseudoinverse:
Another method for computing the pseudoinverse uses the recursion
We now show that formula_6 is a pseudoinverse of formula_1:
Note that formula_9 is the pseudoinverse of formula_10.
We now show that formula_16 is a pseudoinverse of formula_15:
where is the pseudoinverse of , which is formed by replacing every non-zero diagonal entry by its reciprocal and transposing the resulting matrix. The pseudoinverse is one way to solve linear least squares problems.
The pseudoinverse of the null (all zero) vector is the transposed null vector. The pseudoinverse 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 pseudoinverse, i.e., and , where and denote the pseudoinverse of "A" and "B".
In mathematics, block matrix pseudoinverse is a formula of pseudoinverse 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 pseudoinverse for formula_91 by verifying that the defining properties of the pseudoinverse hold, when formula_13 is Hermitian and idempotent.
The MASS package for R provides a calculation of the Moore–Penrose pseudoinverse through the codice_5 function. The codice_5 function calculates a pseudoinverse using the singular value decomposition provided by the codice_7 function in the base R package.
It is also possible to define a pseudoinverse for scalars and vectors. This amounts to treating these as matrices. The pseudoinverse of a scalar is zero if is zero and the reciprocal of otherwise:
where formula_6 is the Moore-Penrose pseudoinverse of "A" and "w" is any "n"×1 vector.
The pseudoinverse 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 pseudoinverse:
For the case where is not invertible the Moore–Penrose pseudoinverse should be used instead.
This particular pseudoinverse constitutes a "left inverse", since, in this case, formula_34.