Which Matrix Norms Are Invariant Under Change of Basis?

[Heimdall]

Hi,

I don't get which of the many matrix norms is invariant through a change of basis. I get that the Frobenius norm is, because it can be expressed as a function of the eigenvalues only. Are there others of such kind of invariant norms?

Thanks

 

[NegativeDept]

By change of basis, I assume you mean changing from one orthonormal basis to another orthonormal basis. If so, then I think any matrix norm which is compatible with the norm you're using for vectors will be the same in either basis.

An important example is the spectral norm. This norm is induced by the Euclidean norm, which is just the usual way of defining "magnitude" for a real 2D or 3D vector. Roughly speaking, the spectral norm is the maximum amount that a matrix can "stretch" a vector.