Definition:Moore-Penrose Pseudoinverse
Definition
Let $\mathbf A$ be a matrix of order $m \times n$ over the complex numbers $\C$.
Definition 1
The Moore-Penrose pseudoinverse of $\mathbf A$ is the unique $n \times m$ matrix $\mathbf A^+$ defined as:
- $\mathbf A^+ = \begin {cases} \paren {\mathbf A^\intercal \mathbf A}^{-1} \mathbf A^\intercal & : m > n \\ \mathbf A^\intercal \paren {\mathbf A \mathbf A^\intercal}^{-1} & : n < m \end {cases}$
where $\mathbf A^\intercal$ denotes the transpose of $\mathbf A$.
Definition 2
The Moore-Penrose pseudoinverse of $\mathbf A$ is the unique $n \times m$ matrix $A^+$ which satisfies the four Moore-Penrose conditions:
| \((1)\) | $:$ | \(\ds \mathbf A \mathbf X \mathbf A \) | \(\ds = \) | \(\ds \mathbf A \) | |||||
| \((2)\) | $:$ | \(\ds \mathbf X \mathbf A \mathbf X \) | \(\ds = \) | \(\ds \mathbf X \) | |||||
| \((3)\) | $:$ | \(\ds \mathbf A \mathbf X \) | \(\ds = \) | \(\ds \paren {\mathbf A \mathbf X}^* \) | |||||
| \((4)\) | $:$ | \(\ds \mathbf X \mathbf A \) | \(\ds = \) | \(\ds \paren {\mathbf X \mathbf A}^* \) |
where $\mathbf A^*$ denotes the Hermitian conjugate of $\mathbf A$.
Examples
Arbitrary Example
Let $\mathbf A$ be the matrix:
- $\mathbf A := \begin {pmatrix} 2/3 & 0 \\ 5/6 & 1/2 \\ 1/3 & 1 \end {pmatrix}$
The Moore-Penrose pseudoinverse of $\mathbf A$ is:
- $\mathbf A^+ := \begin {pmatrix} 5/6 & 2/3 & -1/3 \\ -1/2 & 0 & 1 \end {pmatrix}$
Also known as
A Moore-Penrose pseudoinverse can also be seen hyphenated: Moore-Penrose pseudo-inverse.
Some sources refer to it as a Moore-Penrose inverse.
Some sources refer to it as just a pseudoinverse, or again hyphenated: pseudo-inverse
Also see
- Results about Moore-Penrose pseudoinverses can be found here.
Source of Name
This entry was named for Eliakim Hastings Moore and Roger Penrose.
Historical Note
The concept of the Moore-Penrose pseudoinverse was defined by Eliakim Hastings Moore in $1920$, and then again by Roger Penrose in $1955$.
Sources
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Moore-Penrose conditions, Moore-Penrose pseudoinverse