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A class of matrix norms, denoted $\Vert\cdot\Vert_p$ is defined as
The matrix $p$ norms are defined in terms of the vector $p$ norms.
An alternate definition is
As with vector $p$ norms, the most important are the 1, 2, and $\infty$ norms. The 1 and $\infty$ norms are very easy to calculate for an arbitrary matrix:
It directly follows from this that $\Vert\,A\,\Vert_1 = \Vert\,A^T\,\Vert_\infty$
The calculation of the $2$ norm is more complicated. However, it can be shown that the 2-norm of $A$ is the square root of the largest eigenvalue of $A^TA$ There are also various inequalities that allow one to make estimates on the value of $\Vert\,A\,\Vert_2$
\begin{equation*} \frac{1}{\sqrt{n}}\Vert\,A\,\Vert_\infty \leq \Vert\,A\,\Vert_2 \leq \sqrt{m}\Vert\,A\,\Vert_\infty. \end{equation*} \begin{equation*} \frac{1}{\sqrt{m}}\Vert\,A\,\Vert_1 \leq \Vert\,A\,\Vert_2 \leq \sqrt{n}\Vert\,A\,\Vert_1 . \end{equation*} \begin{equation*} \Vert\,A\,\Vert_2^2\leq\Vert\,A\,\Vert_\infty\cdot\Vert\,A\,\Vert_1. \end{equation*} \begin{equation*} \Vert\,A\,\Vert_2 \leq \Vert\,A\,\Vert_F\leq\sqrt{n}\Vert\,A\,\Vert_2. \end{equation*} ($\Vert\,A\,\Vert_F$ is the Frobenius matrix norm)
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