Our text introduced matrix norms in the context of proving the second order conditions for max and min. But the concept is more generally useful, even for nonsymmetric matrices, so I wrote a set of problems to get at the more general situation as well. This problem set gives specific examples of this phenomenon, and also extends student understanding of the related theory in some other ways. This problem set gives concrete examples. The Hessian matrix of 2nd-order partials is almost always symmetric.
Symmetric matrices are diagonalizable with real, orthogonal eigenvalues.
In this problem set they work out the proofs. The concepts and notation of Big O and little o provide a wonderful way to clear your paper of convergence details, which details can be especially nasty in several variables. If you work through this long, sequenced problem set, you should come to understand this notation very well, and become accomplished at using it. This problem set starts with definitions of the implicit, explicit, and parametric geometric representation of functions, and goes on to problems that also bring in the tangents spaces to these representations. Most texts state the inverse and implicit function theorems, and honors texts prove them.