462: Frobenius Matrix Norm

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A definition of Frobenius matrix norm

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Topics

About: matrix

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Definition
• 2: Note

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Starting Context

• The reader knows a definition of matrix.

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Target Context

• The reader will have a definition of Frobenius matrix norm.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Definition

For any real or complex matrix, $M$, $\sqrt{\sum _{j,k}|M_{j,k}{|}^2}$, denoted as $\Vert M{\Vert }_F$

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2: Note

By the equivalence of norms for finite vectors space theorem, for any finite-dimensional matrix norm, $\Vert \bullet \Vert$, $r_1\Vert M{\Vert }_F\le \Vert M\Vert \le r_2\Vert M{\Vert }_F$ for some positive numbers, $r_1,r_2\in \mathbb{R}$, which can be used to evaluate $\Vert M\Vert$, while what exactly $r_1,r_2$ are does not matter in many cases.

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References

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