Question

Explain why $\|A\|=\max \left|a_{i j}\right|$ defines a norm on the space of $n \times n$ matrices. Show by example that this is not a matrix norm, i.e., (3.42) is not necessarily valid.

    Explain why $\|A\|=\max \left|a_{i j}\right|$ defines a norm on the space of $n \times n$ matrices. Show by example that this is not a matrix norm, i.e., (3.42) is not necessarily valid.

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 3, Problem 52

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Step 1

The norm defined is $\|A\| = \max |a_{ij}|$, where $A = (a_{ij})$ is an $n \times n$ matrix. To verify that this is a norm, we need to check the following properties: - Non-negativity: $\|A\| \geq 0$. - Definiteness: $\|A\| = 0$ if and only if $A = 0$. - Show more…

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Explain why $\|A\|=\max \left|a_{i j}\right|$ defines a norm on the space of $n \times n$ matrices. Show by example that this is not a matrix norm, i.e., (3.42) is not necessarily valid.

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