L2 Norm of Pseudo-Inverse Relation with Minimum Singular Value

Hint: It suffices to prove the following facts:

  • If $U,V$ are orthogonal (and square), then $\|AV\| = \|UA\| = \|A\|$
  • $(U\Sigma V^T)^\dagger = V\Sigma^{\dagger}U^T$ (where $\dagger$ denotes the pseudo-inverse)
  • If $\Sigma$ is a diagonal matrix of singular values, then $\|\Sigma^{\dagger}\| = 1/\sigma_{min}(\Sigma)$

Perhaps you can put the pieces together from here. Happily, this approach still works when $A$ does not have full column-rank.

Tags:

Linear Algebra

Norm

Matrices

Pseudoinverse

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