Eritque Arcus Math Notes

pseudo inverse (Moore-Penrose Inverse)

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A no-square π‘šΓ—π‘› matrix 𝐴 usually has no inverse.
By Rank of matrix

  • if π‘Ÿπ‘Žπ‘›π‘˜(𝐴)=𝑛 we can say 𝐴+=(𝐴𝑇𝐴)βˆ’1𝐴𝑇 and π‘π‘œπ‘›π‘‘(𝐴)=‖𝐴𝑇‖2⋅‖𝐴+β€–2.
  • if π‘Ÿπ‘Žπ‘›π‘˜(𝐴)<𝑛, we can say π‘π‘œπ‘›π‘‘(𝐴)=∞.
    A simple construction method by Singular value decomposition is
𝐴+=𝑉Σ+π‘ˆπ‘‡

where Ξ£+ is (βˆ€π›ΏβˆˆΞ£>0)𝛿+=1𝛿.

Applications

Replacing normal equation to solve polynomial when (𝐴𝑇𝐴) doesn’t have inverse: least squares solution of 𝐴π‘₯β‰ˆπ‘ is given by π‘₯=𝐴+𝑏.

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