View source for Moore–Penrose pseudoinverse

{{Wikipedia| Moore–Penrose pseudoinverse }}
The '''Moore–Penrose pseudoinverse''', denoted ''A''{{+}}, is a generalization of the {{w|invertible matrix|inverse matrix}} that can be used to compute {{w|least squares}} solutions for overdetermined {{w|system of linear equations|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, {{nowrap|''A''{{+}} {{=}} ''A''{{inv}}}}
* If ''A'' has rational entries, so does ''A''{{+}}
* {{nowrap|(''A''{{+}}){{+}} {{=}} ''A''}}
* {{nowrap|(''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''
* {{nowrap|''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 {{nowrap|''A''{{+}} {{=}} ''A''{{t}}(''AA''{{+}}){{inv}}}}. 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_for_.5Bmath.5D.F0.9D.91.9D.3D2.5B.2Fmath.5D|Dave Keenan & Douglas Blumeyer's guide to RTT/Tuning computation#Pseudoinverse method]]
* [[Generator embedding optimization #Pseudoinverse: the "how"]]
* [[Generator embedding optimization #Pseudoinverse: the "why"]]

[[Category:Math]]