Moore-Penrose Inverse¶

Definition¶

Let $A$ be a real $m\times n$ matrix. The Moore-Penrose inverse of $A$ is the $n\times m$ matrix $A^+$ satisfying the four Penrose conditions

\text{(1) }AA^+A=A,\quad \text{(2) }A^+AA^+=A^+,\quad \text{(3) }(AA^+)^{\top}=AA^+,\quad \text{(4) }(A^+A)^{\top}=A^+A.

\begin{aligned} &\text{(1) }AA^+A=A,\\ &\text{(2) }A^+AA^+=A^+,\\ &\text{(3) }(AA^+)^{\top}=AA^+,\\ &\text{(4) }(A^+A)^{\top}=A^+A. \end{aligned}

For every $A$ exactly one such matrix exists.

Interpretation¶

Condition (1) alone defines a generalized inverse, of which there are usually many. Condition (2) makes $A$ in turn a generalized inverse of $A^+$ , and conditions (3) and (4) require the two products to be symmetric. Together they force $AA^+$ and $A^+A$ to be orthogonal projections, which pins down a single matrix.

The four fundamental subspaces as boxes: A maps the row space to the column space and A+ maps it back, while Null(A) and Null(A transpose) are sent to zero.

In the figure, $A$ (blue) carries the row space $\operatorname{Row}(A)$ onto the column space $\operatorname{Col}(A)$ and sends $\operatorname{Null}(A)$ to $\boldsymbol{0}$ . The Moore-Penrose inverse $A^+$ (red) runs the other way: it undoes $A$ from $\operatorname{Col}(A)$ back onto the row space and sends $\operatorname{Null}(A^{\top})$ to $\boldsymbol{0}$ . This is why $AA^+$ projects onto $\operatorname{Col}(A)$ and $A^+A$ onto $\operatorname{Row}(A)$ .

Geometrically, $A^+$ inverts $A$ where it can and sends the rest to zero: it maps $\operatorname{Col}(A)$ back onto the row space by undoing $A$ , and maps $\operatorname{Null}(A^{\top})$ to $\boldsymbol{0}$ .

Two cautions. In general $(AB)^+\neq B^+A^+$ $B^{+} A^{+}$ . And $A^+$ does not depend continuously on $A$ : for $\varepsilon\neq0$ $0$ , $\operatorname{diag}(1,\varepsilon)^+=\operatorname{diag}(1,1/\varepsilon)$ $diag (1, 1/ ε)$ , which explodes as $\varepsilon\to0$ $0$ , while $\operatorname{diag}(1,0)^+=\operatorname{diag}(1,0)$ $diag (1, 0)$ .

Key Results¶

Via the singular value decomposition $A=U\Sigma V^{\top}$ $U Σ V^{⊤}$ : $A^+=V\Sigma^+U^{\top}$ $V Σ^{+} U^{⊤}$ , where the $n\times m$ matrix $\Sigma^+$ has $1/\sigma_i$ in place of each positive singular value $\sigma_i$ and zeros elsewhere.
If $A$ is square and invertible, $A^+=A^{-1}$ $A^{- 1}$ . For a scalar, $a^+=1/a$ if $a\neq0$ and $0^+=0$ $0$ .
Full column rank: $A^+=(A^{\top}A)^{-1}A^{\top}$ $(A^{⊤} A)^{- 1} A^{⊤}$ , a left inverse. Full row rank: $A^+=A^{\top}(AA^{\top})^{-1}$ $A^{⊤} (A A^{⊤})^{- 1}$ , a right inverse.
$AA^+$ is the orthogonal projection onto $\operatorname{Col}(A)$ and $A^+A$ the orthogonal projection onto the row space $\operatorname{Row}(A)$ .
$\boldsymbol{x}=A^+\boldsymbol{b}$ is the least-squares solution of $A\boldsymbol{x}=\boldsymbol{b}$ of minimum norm; if the system is consistent, it is its minimum-norm solution.
$(A^+)^+=A$ $A$ , $(A^{\top})^+=(A^+)^{\top}$ $(A^{+})^{⊤}$ , $\operatorname{rank}A^+=\operatorname{rank}A$ $rank A$ .

Examples¶

A singular matrix. Let $A=\begin{pmatrix}1&1\\1&1\end{pmatrix}=2\boldsymbol{u}\boldsymbol{u}^{\top}$ with $\boldsymbol{u}=\tfrac{1}{\sqrt{2}}(1,1)^{\top}$ $\frac{1}{2} (1, 1)^{⊤}$ , so its only positive singular value is $2$ . Then

A^+=\tfrac{1}{2}\boldsymbol{u}\boldsymbol{u}^{\top}=\tfrac{1}{4}\begin{pmatrix}1&1\\1&1\end{pmatrix}.

Check: $A^2=2A$ $2 A$ , so $AA^+=A^+A=\tfrac{1}{2}A$ $\frac{1}{2} A$ , which is symmetric, and $AA^+A=\tfrac{1}{2}A^2=A$ $A$ , $A^+AA^+=\tfrac{1}{8}A^2=\tfrac{1}{4}A=A^+$ $A^{+}$ . By contrast the generalized inverse $G=\begin{pmatrix}1&0\\0&0\end{pmatrix}$ gives $AG=\begin{pmatrix}1&0\\1&0\end{pmatrix}$ $(1100)$ , which is not symmetric, so condition (3) fails.

A single column. For $\boldsymbol{a}=(1,2)^{\top}$ (full column rank), $\boldsymbol{a}^+=(\boldsymbol{a}^{\top}\boldsymbol{a})^{-1}\boldsymbol{a}^{\top}=\tfrac{1}{5}(1,2)$ $\frac{1}{5} (1, 2)$ . For $\boldsymbol{b}=(1,1)^{\top}$ the least-squares solution of $\boldsymbol{a}x=\boldsymbol{b}$ is $x=\boldsymbol{a}^+\boldsymbol{b}=\tfrac{3}{5}$ $\frac{3}{5}$ .

Prerequisites¶

Sources¶

Math V: Advanced Linear Algebra · Week 7

Aliases: pseudoinverse, Moore-Penrose pseudoinverse, pseudo-inverse, MP inverse, Moore-Penrose generalized inverse Keywords: A^+, Penrose conditions, A^+ = V Σ^+ U^T, (A^T A)^{-1} A^T, minimum-norm least-squares solution Symbols: epsilon, sigma, square root

Last updated: 17 October 2026 21:53