461: Matrix Norm Induced by Vector Norms

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A definition of matrix norm induced by vector norms

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Topics

About: vectors space

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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 normed vectors space.
• The reader knows a definition of matrix.

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Target Context

• The reader will have a definition of matrix norm induced by vector norms.

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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 normed real or complex column vectors spaces, $V_1,V_2$, and any real or complex matrix, $M$, as the representative of any linear map from $V_1$ into $V_2$, $sup\{\Vert Mv\Vert /\Vert v\Vert |v\in V_1,\Vert v\Vert \ne 0\}$, denoted as $\Vert M\Vert$

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2: Note

As the definition depends on the norms of $V_1,V_2$, it is meaningless to talk about the matrix norm in this sense without specifying the normed vectors spaces.

Although it is not clear what exactly the norm is for a matrix, for any finite-dimensional matrix, $M$, by the equivalence of norms for finite vectors space theorem, $r_1\Vert M{\Vert }_F\le \Vert M\Vert \le r_2\Vert M{\Vert }_F$ for some positive numbers, $r_1,r_2\in \mathbb{R}$, where $\Vert \bullet {\Vert }_F$ denotes the Frobenius norm, and $\Vert Mv\Vert \le \Vert M\Vert \Vert v\Vert \le r_2\Vert M{\Vert }_F\Vert v\Vert$, which is enough for many cases even with $r_2$ unknown. Especially, $\Vert M\Vert$ is always finite.

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References

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