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What's geometric interpretation of the Euclidean (Frobenius) norm of the matrix?

I know that vector norm is a vector's lenght but what is the matrix norm geometrically?

Rodrigo de Azevedo
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DKay
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  • Consider the matrix as the operator on your vector space. Imagine the unit ball in this space; apply your operator to that unit ball; the lenght of the longest vector in the image of the unit ball will be the norm of the operator – Anton Apr 17 '20 at 11:05
  • I heard its an iterpretation for m-norm. Is it works with euclidian one too? – DKay Apr 17 '20 at 11:20
  • this interpretation doesn't really depends on which norm you choose, because it's based on general defenition of operator-norm – Anton Apr 17 '20 at 11:56

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Answer was given by Anton Zagrivin.

Consider the matrix as the operator on your vector space. Imagine the unit ball in this space; apply your operator to that unit ball; the lenght of the longest vector in the image of the unit ball will be the norm of the operator

DKay
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    This is a good description of the operator norm, but does not apply to the Frobenius norm, which is not an operator norm. This answer is a good geometric description of the Frobenius norm. – Danica Sep 04 '21 at 21:09