The Frobenius norm of a matrix, calculated as 1.45 in this discussion, does not directly indicate whether a matrix is well-posed or ill-posed. The critical factor for determining matrix conditioning is the ratio of the largest singular value to the smallest singular value. A significant disparity, such as a ratio of 10^40, indicates an ill-conditioned matrix, despite the Frobenius norm being relatively low at 1.5. This highlights the importance of understanding the relationship between different norms and their implications for numeric stability.
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StoneTemplePython said: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.