33:
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683:
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530:
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227:
110:
850:
47:
53:
691:
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558:). This means that one needs to use a generalized inverse for the solution of the constrained problem. So, a generalized inverse of
85:
with an additional constraint that the solution is in a given subspace. One also says that the problem is described by a system of
363:
82:
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196:{\displaystyle Ax=b\qquad ({\text{with given }}A\in \mathbb {R} ^{m\times n}{\text{ and }}b\in \mathbb {R} ^{m})}
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An example of a pseudoinverse that can be used for the solution of a constrained problem is the
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has a solution if and only if the unconstrained system of equations
787:{\displaystyle A_{L}^{(-1)}:=P_{L}(AP_{L}+P_{L^{\perp }})^{-1},}
26:
532:
of the constrained problem is invertible (in that case,
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417:{\displaystyle (AP_{L})x=b\qquad x\in \mathbb {R} ^{n}}
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This article contains obvious mistakes, as noted in
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48:Talk:Constrained generalized inverse#Dimensions
797:if the inverse on the right-hand-side exists.
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42:needs attention from an expert in Mathematics
8:
206:is acceptable only when it is in a certain
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512:may be singular even if the system matrix
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312:. Constrained system of linear equations
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92:In many practical problems, the solution
56:may be able to help recruit an expert.
7:
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685:, which is defined by the equation
823:. You can help Knowledge (XXG) by
25:
347:{\displaystyle Ax=b\qquad x\in L}
807:
469:{\displaystyle \mathbb {R} ^{n}}
251:{\displaystyle \mathbb {R} ^{n}}
112:of a linear system of equations
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79:constrained generalized inverse
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427:is solvable. If the subspace
87:constrained linear equations
44:. The specific problem is:
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83:system of linear equations
616:constrained pseudoinverse
81:is obtained by solving a
587:{\displaystyle (AP_{L})}
505:{\displaystyle (AP_{L})}
447:is a proper subspace of
54:WikiProject Mathematics
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305:{\displaystyle P_{L}}
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263:orthogonal projection
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643:Bott–Duffin inverse
551:{\displaystyle m=n}
18:Bott–Duffin inverse
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678:{\displaystyle L}
658:{\displaystyle A}
631:{\displaystyle A}
607:{\displaystyle L}
594:is also called a
525:{\displaystyle A}
440:{\displaystyle L}
278:{\displaystyle L}
222:{\displaystyle L}
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105:{\displaystyle x}
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16:(Redirected from
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75:linear algebra
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875:Matrix stubs
825:expanding it
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62:October 2019
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41:
864:Categories
774:−
762:⊥
709:−
400:∈
339:∈
176:∈
160:×
147:∈
870:Matrices
817:matrices
819:is a
821:stub
77:, a
645:of
618:of
265:on
229:of
73:In
866::
720::=
638:.
258:.
89:.
852:e
845:t
838:v
827:.
782:,
777:1
770:)
758:L
753:P
749:+
744:L
740:P
736:A
733:(
728:L
724:P
715:)
712:1
706:(
701:L
697:A
673:L
653:A
626:A
614:-
602:L
582:)
577:L
573:P
569:A
566:(
546:n
543:=
540:m
520:A
500:)
495:L
491:P
487:A
484:(
462:n
457:R
435:L
410:n
405:R
397:x
393:b
390:=
387:x
384:)
379:L
375:P
371:A
368:(
342:L
336:x
332:b
329:=
326:x
323:A
298:L
294:P
273:L
244:n
239:R
217:L
191:)
186:m
181:R
173:b
163:n
157:m
152:R
144:A
136:(
132:b
129:=
126:x
123:A
100:x
64:)
60:(
50:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
