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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.

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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.

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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.

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how about the second part in my question? – Salih Ucan Jan 3 '13 at 16:35

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