Pseudo-inverse of a matrix

Alicia Y. Tsai; Authors: Laurent EI Ghaoui; Contributors: Phan Hoang, Tony Tin, Ha To Tam, Nguyen Van Kep, Le Dinh Nam, Juliette Decugis, Nguyen Quoc Cuong.; Giuseppe C. Calafiore

Pseudo-inverse of a matrix

The pseudo-inverse of a $m \times n$ matrix $A$ is a matrix that generalizes to arbitrary matrices the notion of inverse of a square, invertible matrix. The pseudo-inverse can be expressed from the singular value decomposition (SVD) of $A$ , as follows.

Let the SVD of $A$ be

$A = U \left( \begin{array}{cc} S & 0 \\ 0 & 0 \end{array} \right) V^T,$

where $U,V$ are both orthogonal matrices, and $S$ is a diagonal matrix containing the (positive) singular values of $A$ on its diagonal.

Then the pseudo-inverse of $A$ is the $n \times m$ matrix defined as

$A^\dagger = V \left( \begin{array}{cc} S^{-1} & 0 \\ 0 & 0 \end{array} \right) U^T.$

Note that $A^\dagger$ has the same dimension as the transpose of $A$ .

This matrix has many useful properties:

● If $A$ is full column rank, meaning $\mathbf{rank}(A) = n \leq m$ , that is, $A^TA$ is not singular, then $A^\dagger$ is a left inverse of $A$ , in the sense that $A^\dagger A = I_n$ . We have the closed form expression

$A^\dagger = (A^TA)^{-1}A^T.$

● If $A$ is full row rank, meaning $\mathbf{rank}(A) = m \leq n$ , that is, $AA^T$ is not singular, then $A^\dagger$ is a right inverse of $A$ , in the sense that $AA^\dagger = I_m$ . We have the closed-form expression