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Exercise 2.2.11

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If $Q^{T} Q = I$ , let $Q = U Σ V^{T}$ , then we have $Q^{+} = V Σ^{+} U^{T}$ , $Q^{T} Q = V Σ U^{T} U Σ V^{T} = V Σ^{2} V^{T} = I$ , multiply left by $V^{T}$ and right side by $V$ , we have $Σ^{2} = I$ , so $Σ = I$ , thus $Σ^{+} = I$ and $Q^{+} = Q^{T}$ .

If $A = QR$ for invertible $R$ , the we see that $A^{+} = R^{- 1} Q^{+}$ where $A A^{+} = I$ , so $A A^{+} = QR R^{- 1} Q^{+} = Q Q^{+} = Q Q^{T}$ .

niuers

2020-03-20 00:00

Exercise 2.2.11

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