Oak asked in Science & MathematicsMathematics · 1 decade ago

Pllllllllleeeeeeeeaaaaaaasssssseeeeee help!!! Proof involving moore-penrose pseudo inverse!!!?

If A^+ denotes the moore penrose pseudoinverse of A could someone please prove (A*A)^+=(A^+)(A*)^+ . This is really important so i would really appreciate any help. Naturally 10 points will be awarded to the best answer!

Thank you

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  • 1 decade ago
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    It's been a while since I've done matrix work, and I never got to pseudoinverses, so the following, in all probability, (a) is longer than it needs to be; and (b) should be checked by you for correctness. Still, even if it's not satisfactory, it may be enough for you to get where you want to go.

    I make use of identities whose short proofs are given at the Wikipedia link below. Note also that conjugate transpose (*) commutes with pseudoinverse (+).

    For any A, A+ = A+ (A*)+ A* and (A+)* = A A+ (A*)+. We can multiply these together to get

    A+ (A*)+ = A (A+)* = A+ (A*)+ A* A A+ (A*)+

    We also have A* = A* A A+ and A = (A*)+ A* A. And again, multiplying, we have

    A* A = A* A A+ (A*)+ A* A

    *****

    Next,

    A* A A+ (A*)+ = A* (A*)+ = A+ A A* (A*)+ = A+ A = A+ (A+)* A* A = (A* A A+ (A*)+)*

    and

    A+ (A*)+ A* A = A+ A = A+ A A* (A*)+ = A* (A*)+ = A* A A+ (A*)+ = (A+ (A*)+ A* A)*

    *****

    Now, let U = (A* A) and V = A+ (A*)+. We have just shown that

    V = V U V

    U = U V U

    U V = (U V)*

    V U = (V U)*

    Since for any U, U+ exists and is unique, and V satisfies the

    requirements of U+, V = U+.

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