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Possible Duplicate:
Subordinate matrix norm

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

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.

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.

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    @rookieRailer Please do not delete the question, just let it be. (In any case, since there is already an answer with upvotes, I think you will not be allowed to delete it now.)2012-01-11

1 Answers 1

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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.

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    Very nice, Chris!2012-01-10