Equivalence of matrix norms

Question

The equivalence of vector norms on finite dimensional spaces immediately implies that all induced matrix norms are equivalent.

However, for matrix norms (like Frobenius norm, Nuclear norm) that are not induced by a vector norm, one proves equivalence on a case by case basis.

Hence my question is, can I say all matrix norms, induced or not, are equivalent ? If so how do I justify it in general ?

Answer 1

The wiki article on matrix norms summarizes it very well:

In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).

So a matrix norm is always a vector norm - on the (vector) space of $n\times n$ matrices. Since this space is finite dimensional, all vector norms on it are equivalent. And because equivalence of vector norms is the same condition as equivalence of matrix norms, all matrix norms are equivalent too.

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You say,

However, for matrix norms (like Frobenius norm, Nuclear norm) that are not induced by a vector norm, one proves equivalence on a case by case basis.

Which is true if you want to compute good estimates on the coefficients $$a\|\cdot\|_{\alpha}\le\|\cdot\|_{\beta}\le b\|\cdot\|_{\alpha}.$$

But since any matrix norm - induced or not - satisfies the properties of a vector norm, the theorem of vector norm equivalence (asserting only existence of the coefficients) applies. The describing properties imply equivalence (in finite dimensions), not the particular norm definition.