Let left bracketing bar z right bracketing bar be a vector norm of a vector z such that left double bracketing bar A right double bracketing bar = max_( left bracketing bar z right bracketing bar = 1) left double bracketing bar A z right double bracketing bar . Then left double bracketing bar A right double bracketing bar is a matrix norm which is said to be the natural norm induced (or subordinate) to the vector norm left bracketing bar z right bracketing bar . For any natural norm, left double bracketing bar I right double bracketing bar = 1, where I is the identity matrix. The natural matrix norms induced by the L^1-norm, L^2-norm, and L^∞-norm are called the maximum absolute column sum norm, spectral norm, and maximum absolute row sum norm, respectively.
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