Pseudoinverse
Pseudoinverse
A generalized inverse for any matrix can be defined using URV factorization or SVD.
Given a URV factorization of matrix \mathbf{A} \in \mathbb{R}^{m \times n}
\mathbf{A} = \mathbf{U} \begin{bmatrix} \mathbf{C} & 0 \\ 0 & 0 \\ \end{bmatrix} \mathbf{V}^{T}
the pseudoinverse of \mathbf{A} is defined as
\mathbf{A}^{\dagger} = \mathbf{V} \begin{bmatrix} \mathbf{C}^{-1} & 0 \\ 0 & 0 \\ \end{bmatrix} \mathbf{U}^{T}.
The pseudoinverse of matrix \mathbf{A} can also be stated as a matrix \mathbf{A}^{\dagger} that satisfies the following:
\mathbf{A} \mathbf{A}^{\dagger} \mathbf{A} = \mathbf{A},
\mathbf{A}^{\dagger} \mathbf{A} \mathbf{A}^{\dagger} = \mathbf{A}^{\dagger},
where \mathbf{A} \mathbf{A}^{\dagger} and \mathbf{A}^{\dagger} \mathbf{A} are symmetric matrix:
(\mathbf{A} \mathbf{A}^{\dagger})^{T} = \mathbf{A} \mathbf{A}^{\dagger},
(\mathbf{A}^{\dagger} \mathbf{A})^{T} = \mathbf{A}^{\dagger} \mathbf{A}.
Properties of pseudoinverse
The pseudoinverse \mathbf{A}^{\dagger} of \mathbf{A} is unique.
Given a full rank matrix \mathbf{A} \in \mathbb{R}^{m \times n}.
If m > n, the left inverse of \mathbf{A} is the pseudoinverse of \mathbf{A} and is written as
(\mathbf{A}^{T} \mathbf{A})^{-1} \mathbf{A}^{T}.
If m < n, the right inverse of \mathbf{A} is the pseudoinverse of \mathbf{A} and is written as
\mathbf{A}^{T} (\mathbf{A} \mathbf{A}^{T})^{-1} .
If m = n, the inverse of \mathbf{A} is the pseudoinverse of \mathbf{A}.