Thursday, 6 March 2014

linear algebra - Least-squares left-inverse having smallest Frobenius norm

While trying to prove that the left-inverse of A provided by the least-squares solution to y=Ax has the smallest Frobenius norm, I am stuck at a point which I describe below:



Let B be any left-inverse of a full-rank tall matrix A, i.e., BA=I. Let the QR-decomposition: A=QR. In this case, R is invertible since A is full-rank and Q has orthonormal columns as always.



I want to show that . Any ideas? The rest of the proof to show that the least-squares left-inverse has the smallest Frobenius norm is in place and I will be done if I can show this.

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