Recently I started exploring convergence of some iterative methods and spotted the equivalent of the spectral radius and a matrix norm.
For instance, http://www.scribd.com/doc/37323755/36/Richardson-Iteration states in Example 1.28 that 2-norm of a matrix is its spectral radius. On the other hand, What is the difference between the Frobenius norm and the 2-norm of a matrix? states a difference.
What is the difference between the Frobenius and 2-norm of a matrix? Is the class of symmetric matrices for which the equality of 2-norm and the spectral radius holds?
Best Answer
OK. Regarding your first question, the difference I see is that Frobenius norm is a matrix norm while a matrix 2-norm is induced by the vector 2-norm, i.e., $\|A\|_2=\max_{\|x\|=1}\|Ax\|_2$. In fact, $\|A\|_2$ is the maximal singular value of $A$, that is, the square root of the maximal eingenvalue of $A^TA$ (this more computable). This also answer your second question. In general the spectral radius of a matrix is less or equal than the matrix norm.