Homework 3 Task 10: Frobenius Norm and Consistent Matrix Norms
Task: Search the internet for sites that define and discuss the Frobenius matrix norm. Also, look for sites that define consistent matrix norms. Summarize in a paragraph or two what you found and as usual cite the sites.
The Frobenius Norm
There were many sites that talked about how to define the Frobenius norm, but I was more curious about some of the merits of using the Frobenius norm, which is why I chose to talk about this Quora forum entry (though I recognize that forums may or may not be a great place to get information). Specifically, the answer by Garrett Thomas explains that one of the values of using the Frobenius norm is that it is useful for gradient-based methods, since the Frobenius norm is differentiable with respect to the individual entries of the matrix.
Consistent Norms
This slideshow goes through the definition for consistent matrix norms. The definition that it gives is Consistent Matrix Norms. A submultiplicative matrix norm which is defined for all m, n ∈ N, is said to be a consistent matrix norm. A submultiplicative matrix norm means that a matrix norm of a multiplication of two matrices should be bounded by the product of the norms of the two matrices, i.e.
To summarize, a consistent matrix norm is one which satisfies the above inequality for all sizes of matrices.