Proofs of matrix norm properties

Equality occurs when A and B share a support vector. 

It is also possible for other measures of matrix size to be defined, and they are also called norms provided that they satisfy this additional property as well as the three above. 

All the above applies to matrices which are not square, but if a matrix A is square, then because, if V is an eigencolumn of the matrix, we know that AV = XV, any eigenvalue gives a lower bound for any of the norms. 

Clearly the magnitude of the largest eigenvalue is a tighter lower bound than smaller ones. 

It is equally true that any norm is an upper bound for the largest eigenvalue. 

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