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 🔗

• 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 §linear_regression, giving the linear regression coefficients.
• The pseudoinverse exists and is unique for any matrix $$X$$.

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