Moore–Penrose pseudoinverse: Difference between revisions

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~~See [[~~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}}.

Some of ~~its~~ properties are explained in the following pages:

To name a few of its properties:

* [[Tenney-Euclidean tuning]]

* 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 "how"]]

* [[Generator embedding optimization#Pseudoinverse: the "why"]]

* [[Generator embedding optimization #Pseudoinverse: the "why"]]

[[Category:~~Smart redirect~~]]

[[Category:Math]]

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-See [[Wikipedia: 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}}.
-Some of its properties are explained in the following pages:
+To name a few of its properties:
-* [[Tenney-Euclidean tuning]]
+* 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 "how"]]
-* [[Generator embedding optimization#Pseudoinverse: the "why"]]
+* [[Generator embedding optimization #Pseudoinverse: the "why"]]
-[[Category:Smart redirect]]
+[[Category:Math]]