Matrix norm less than $1$ iteration Is the following true always for a matrix norm 
$$\lVert AB\rVert \leqslant \lVert A\rVert \cdot \lVert B\rVert \text{ ?}$$
Related to this
 given $r$ is positive constant, $H$ is symmetric positive definite
 is the following true :
$$\lVert (rI - H)(rI + H)^{-1}\rVert < 1 $$
or
$(rI - H)(rI + H)^{-1}$  has the spectral radius less than $1$ certainly?
Thank you.
 A: Matrix norms which satisfy the condition (in addition to others) you mentioned are referred to as sub-multiplicative norms. A well-known example of a matrix norm which doesn't satisfy this condition is the max-norm defined as 
\begin{align}
||A||_\max=\max_{i,j}|a_{ij}|
\end{align}
Let $H=UDU^T$ where $U$ is the matrix of eigenvectors and $D$ is the diagonal matrix of positive eigenvalues. Let $[D]_{ii}=d_{i}$ be the $i^{th}$ diagonal entry. Then, using the fact that $UU^T=I$ and that spectral radius is invariant to unitary transformations, we have 
\begin{align}
|| (rI - H)(rI + H)^{-1} || &=|| (rI - D)(rI + D)^{-1} || \\
&=\max_i \left|\frac{r-d_i}{r+d_i}\right|
\end{align}
Then use that
\begin{align}
\max_i \left|\frac{r-d_i}{r+d_i} \right| \leq\max_i \left(\frac{r}{r+d_i}+\frac{d}{r+d_i}\right)=1
\end{align}
Thus the inequalities you ask for should hold. 
A: An online encyclopedia states hat "some (but not all) matrix norms satisfy" this inequality, though some authors include this as part of the definition. SO it depends.
