Is the Frobenius Norm a Reliable Indicator of Matrix Conditioning?

[SeM]

I have calculated that a matrix has a Frobenius norm of 1.45, however I cannot find any text on the web that states whether this is an ill-posed or well-posed indication. Is there a rule for Frobenius norms that directly relates to well- and ill-posed matrices?

Thanks

 

[StoneTemplePython]

I assume by "ill posed" you mean poorly conditioned, i.e. potential numeric stability problems associated with inversion.

Like I said in your thread here:

https://www.physicsforums.com/threads/how-can-i-analyse-and-classify-a-matrix.931056/

it's the ratio of the largest singular value to smallest singular value that matters. I really struggle to imagine why you'd take that information and then ask a question about about adding up all the squared singular values, and asking what that sum (or the square root of that sum) tells us about condition number.
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Perhaps it'd be prudent for you to make your way through the first 7 chapters of Linear Algebra Done Wrong:

https://www.math.brown.edu/~treil/papers/LADW/LADW.html

It would walk you through many of the fundamentals needed to connect the dots between these different ideas.

 

[SeM]

I assume by "ill posed" you mean poorly conditioned, i.e. potential numeric stability problems associated with inversion.

Like I said in your thread here:

https://www.physicsforums.com/threads/how-can-i-analyse-and-classify-a-matrix.931056/

it's the ratio of the largest singular value to smallest singular value that matters. I really struggle to imagine why you'd take that information and then ask a question about about adding up all the squared singular values, and asking what that sum (or the square root of that sum) tells us about condition number.
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Not really need to go through an entire book for this. It's actually as comprehensive as you wrote here, which I was not aware of. However, the strange thing is that if the ratio between the smallest and largest values of the matrix is in the magnitude of 10^40, it defines an ill conditioned matrix, and the Max norm is very high, about 78. However, the Frobenius norm is "only" 1.5 , so I thought it was strange these two norm would deviate so much.