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Mathematics-Online course: Linear Algebra - Normal Forms - Singular Value Decomposition

Moore-Penrose Conditions

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The pseudo inverse (Moore-Penrose inverse) $ A^+$ of an $ m\times n$-matrix $ A$ is uniquely determined by the following identities:

  1. $\displaystyle A A^+ A = A $

  2. $\displaystyle A^+ A A^+ = A^+ $

  3. $\displaystyle (A A^+)^{\operatorname t}= A A^+ $

  4. $\displaystyle (A^+ A)^{\operatorname t}= A^+ A\,. $

(Authors: App/Burkhardt/Höllig)
It can easily be verified that the pseudo inverse defined by

$\displaystyle A^+ = V S^+ U^{\operatorname t},\quad A = U S V^{\operatorname t} $

satisfies the given conditions. The first identity, for example, follows from

$\displaystyle (U S V^{\operatorname t})(V S^+ U^{\operatorname t})(U S V^{\operatorname t}) = U [SS^+S] V^{\operatorname t} \,, $

since the product of the generalised diagonal matrices (in square brackets) is formed by multiplying the diagonal elements, and since, by definition,

$\displaystyle s^+_{i,i} = 1/s_{i,i},\quad i\le \operatorname{Rg}(A)\,. $

The other identities also follow by simple calculation.

However, to show that $ A^+$ is uniquely characterised by the given conditions is much more difficult.

(Authors: App/Burkhardt/Höllig)
  automatically generated 4/21/2005