A pseudoinverse \(X^+\) is a generalization of the inverse matrix. The most common matrix pseudoinverse is the Moore-Penrose inverse.

- When \(X\) has linearly independent columns (\(X^\top X\) is invertible),
- \(X^+ = (X^\top X)^{-1} X^\top\)
- This a left inverse, since \(X^+X = I\)

- When \(X\) has linearly independent rows (\(X X^\top\) is invertible),
- \(X^+ = X^\top (XX^\top)^{-1}\)
- This a left inverse, since \(XX^+ = I\)

## Common applications π

- Compute a “best fit” (least squares) solution to a system of linear equations with non-unique solutions (e.g. under-determined systems).
- Given \(Ax = b\), if \(x\) does not exist or is not unique, the solution with minimum Euclidean norm is \(A^+ b\). This is the analytical solution to Β§linear_regression, giving the linear regression coefficients.
- The pseudoinverse exists and is unique for any matrix \(X\).

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