Moore–Penrose pseudoinverse

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Moore–Penrose pseudoinverse

The Moore–Penrose pseudoinverse, denoted A⁺, is a generalization of the inverse matrix that can be used to compute least squares solutions for overdetermined systems of linear equations.

To name a few of its properties:

• If A is square and invertible, then its pseudoinverse is equal to its inverse; that is, A⁺ = A⁻¹
• If A has rational entries, so does A⁺
• (A⁺)⁺ = A
• (A^T)⁺ = (A⁺)^T, where A^T is the transpose of A
• AA⁺ is the orthogonal projection matrix that maps onto the space spanned by the columns of A
• A⁺A is the orthogonal projection matrix that maps onto the space spanned by the rows of A
• I − A⁺A, where I is the identity matrix, is the orthogonal projection matrix that maps onto the kernel, or null space, of A
• If the rows of A are linearly independent, then A⁺ = A^T(AA⁺)⁻¹. This means the pseudoinverse can be found in this important special case by people who don't have a pseudoinverse routine available by using a matrix inverse routine.
• uA⁺ is the nearest point to u in the subspace spanned by the rows of A; A⁺v is the nearest point to v in the space spanned by the columns of A.

Some of the properties are explained in the following pages:

• Dave Keenan & Douglas Blumeyer's guide to RTT/Tuning computation#Pseudoinverse method
• Pseudoinverse: the "how"
• Pseudoinverse: the "why"