MPI Moore-Penrose inverse
The Moore-Penrose inverse is a generalization of the matrix inverse. While the matrix inverse exists only for square and invertible matrices, the Moore-Penrose inverse can be defined for any matrix. The Moore-Penrose inverse has many applications in different fields such as statistics, optimization, and control theory.
MPI (Moore-Penrose inverse) is a matrix that can be defined for any matrix A, not necessarily square or invertible. The MPI of A is denoted by A+, and is defined as the unique matrix that satisfies the following four properties:
- AA+A=A
- A+A A= A+
- (AA+)*=AA+
- (A+A)*=A+A
where ( )* denotes the matrix transpose.
The first property states that A+ is a left inverse of A. That is, if we multiply A+ with A from the left, we get the identity matrix. Similarly, the second property states that A+ is a right inverse of A. That is, if we multiply A with A+ from the right, we get the identity matrix.
The third and fourth properties are known as the "pseudoinverse" properties. They are important because they allow us to define the MPI in terms of the singular value decomposition (SVD) of A. The SVD of a matrix A is a factorization of A into three matrices: A = UΣV*, where U and V are unitary matrices and Σ is a diagonal matrix with nonnegative diagonal entries called singular values.
The MPI of A can be expressed as:
A+ = VΣ+U*
where Σ+ is the diagonal matrix with entries 1/σi if σi is nonzero, and 0 otherwise. In other words, Σ+ is obtained from Σ by taking the reciprocal of each nonzero singular value.
The MPI has many applications in different fields. In statistics, it is used for linear regression when the design matrix is not full rank. In optimization, it is used to find the minimum norm solution of an underdetermined system of linear equations. In control theory, it is used to design optimal controllers for systems with non-square transfer functions.
One important property of the MPI is that it minimizes the Frobenius norm of the difference between A and its closest matrix of the same rank. That is, if B is a matrix of the same rank as A, then ||A-A+||F ≤ ||A-B||F, where ||.||F denotes the Frobenius norm of a matrix.
Another important property of the MPI is that it is unique. That is, there is only one matrix that satisfies the four properties listed above. This property follows from the fact that the SVD of a matrix is unique up to sign changes of the columns of U and V.
In conclusion, the MPI is a generalization of the matrix inverse that can be defined for any matrix. It has many applications in different fields such as statistics, optimization, and control theory. The MPI can be expressed in terms of the singular value decomposition of a matrix, and it is unique and minimizes the Frobenius norm of the difference between the original matrix and its closest matrix of the same rank.