Wikipedia:Reference desk archive/Mathematics/2006 August 9

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[edit] Let's play with algebra

What other ways can I express this inequality: \| \mathbf{Ad} \|_2 \geq \epsilon \|\mathbf{d} \|_2 where \mathbf{A} is a matrix, \mathbf{d} is a vector, and ε is a positive scalar? Is there something I can say about the relationship of \mathbf{d} to the nullspace of \mathbf{A}? I also tried playing with the triangle inequality but that didn't get me anywhere. Is there a way to express the equation linearly (strangely enough, in the elements of \mathbf{A}) or otherwise well to use as a constraint in an optimization problem in \mathbf{A}?

What if \mathbf{d} is the gradient of a function? I have this idea that if we're projecting a function f(\mathbf{y}) into a new function g(x)=f(\mathbf{Ax}), ensuring that \| \mathbf{A} \nabla f(y) \|_2 \geq \epsilon \| \nabla f(y)\|_2 will ensure that critical points of g(\mathbf{x}) will also be critical points of f(\mathbf{y}), and that generally if we're performing optimization we can make some progress in minimizing f(\mathbf{y}) by performing optimization over g(\mathbf{x}) and then setting \mathbf{y}_*=\mathbf{Ax}_*, then maybe starting again. Any graphical or intuitive understanding of what this inequality would mean or a better one to choose to accomplish that purpose would be helpful.

If I have second order information, would I do better setting \mathbf{d}=\mathbf{H}^{-1}\nabla f(y) (a Newton's method step) where \mathbf{H} is the Hessian matrix or some approximation to it?

I know my question is confusing. Please just answer whatever small parts you can and go off on any tangents you think might be helpful. 18.252.5.40 08:30, 9 August 2006 (UTC)

The obvious first suggestion is to read our article on matrix norms. --KSmrqT 09:25, 9 August 2006 (UTC)