Moore-Penrose generalized inverse
Let A be an m×n matrix with entries in ℂ. The Moore-Penrose generalized inverse, denoted by A†, is an n×m matrix with entries in ℂ, such that
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1.
AA†A=A
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2.
A†AA†=A†
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3.
AA† and A†A are both Hermitian
Remarks
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•
The Moore-Penrose generalized inverse of a given matrix is unique.
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If A† is the Moore-Penrose generalized inverse of A, then (A†)T is the Moore-Penrose generalized inverse of AT.
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If A=BC such that
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(a)
A∈ℂm×n, B∈ℂm×r, and C∈ℂr×n,
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(b)
r=rank(A)=rank(B)=rank(C), then
A†=C∗(CC∗)-1(B∗B)-1B∗.
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(a)
For example, let
A=(11i010). |
Transform A to its row echelon form to get a decomposition of A=BC, where
B=(1101) and C=(10i010). |
It is readily verified that 2=rank(A)=rank(B)=rank(C). So
A†=12(1-102-ii). |
We check that
AA†=I and A†A=12(10i020-i01) |
are both Hermitian. Furthermore, AA†A=A and A†AA†=A†. So, A† is the Moore-Penrose generalized inverse of A.
Title | Moore-Penrose generalized inverse |
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Canonical name | MoorePenroseGeneralizedInverse |
Date of creation | 2013-03-22 14:31:31 |
Last modified on | 2013-03-22 14:31:31 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 8 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 15A09 |
Classification | msc 60J10 |
Synonym | Moore-Penrose pseudoinverse![]() |
Related topic | DrazinInverse |
Related topic | Pseudoinverse |