Norm $\|A\|$ is not induced by any vector norm

Question

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Subordinate matrix norm

I have a question in my homework for Numerical Linear Algebra, which is as follows:

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Show that the norm $$\|A\| = \max \limits_{i, j} |a_{i,j}|$$ on the space of $n \times n$ real matrices is not induced by any vector norm.

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Can anyone help me solve the problem?

This problem was on the homework given to us on our first class, and I've no idea how to solve.

Thanks.

Answer 1

Another approach: if $\| \cdot \|$ were an induced norm, and $A$ were a matrix with $k$ as a real eigenvalue, then we would have $\|A \|\ge |k|$. But the all-ones matrix has norm $1$ and has $n$ as an eigenvalue.