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- Jan 26, 2012

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**Problem**: The norm of a $m\times n$ matrix $A=[a_{ij}]$ is given by the formula

\[\|A\| = \sqrt{\sum_{i=1}^m\sum_{j=1}^na_{ij}^2}.\]

For an $n\times n$ square matrix $A$, show that the value of $r$ that minimizes $\|A-rI\|^2$ is $r=\text{tr}\,(A)/n$, where $\text{tr}\,(A)$ is the trace of $A$ (i.e. the sum of the main diagonal elements of $A$).

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