Let S be a real and nonsingular matrix, and let ||.|| be any norm on R^n. Define ||.||' by ||x||' = ||Sx||. Show that ||.||' is also a norm on R^n.
I know ||S|| = max ||Sx|| from ||x||=1 but I'm not sure if that applies.
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You just need to check the axioms for a norm. Let , and . Then Positive definiteness: (since is a norm) (since is nonsingular, and thus invertible) Positive homogeneity: (for is a norm) Triangle inequality: (again, is a norm)
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