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I am looking for a reference for the following result:

Let $A$ be an $m \times n$ real matrix. Let $\sigma(A)$ be the spectral norm of $A$. If the largest singular value of $A$ is unique, then

$$ \nabla_A \sigma (A) = \mathbf{u} \mathbf{v}^{\top}, $$

where $\mathbf{u}$ and $\mathbf{v}$ denote the first left- and right-singular (column) vectors, respectively.

I am aware of this question, and am not asking about the proof. Could you please let me know references (e.g., textbooks) about this result? Thank you.

Rodrigo de Azevedo
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Pierre
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  • The answers provided in the linked question seem quite good. A textbook will most likely simply reproduce one of them. What's the point? The result is also subject to direct numerical verification, if that boosts your confidence. – greg Aug 25 '21 at 10:48
  • @RodrigodeAzevedo Thank for editing my question. You can find this result, for example, at https://math.stackexchange.com/questions/929434/derivative-of-spectral-norm-of-symmetric-matrix – Pierre Aug 25 '21 at 15:48
  • @greg Yes, I like the answers provided in the linked question, but I just wanted to know the textbook treatment. – Pierre Aug 25 '21 at 15:48
  • @greg Could you please tell me a textbook that reproduces one of the proofs (in the link)? I need a published reference. Thank you. – Pierre Aug 27 '21 at 20:39
  • Try Harville's Matrix Algebra from a Statistician's Perspective in the chapter on eigenvalues near the end of the book. – greg Aug 28 '21 at 01:55

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