 Solution

ID: past-exam-of-the-numerical-linear-algebra-and-continuous-optimisation-course-of-the-university-of-oxford/2010/a/1/b/solution

∥ x ∥_{2} = (x^{T} x)^{1/2} = (i \sum x_{i}^{2})^{1/2}

, the ordinary length from the origin. The induced matrix norm is

∥ A ∥_{2} = x \neq = 0 sup \frac{∥ A x ∥ _{2}}{∥ x ∥ _{2}} = ∥ x ∥_{2} = 1 max ∥ A x ∥_{2} .

Geometrically this is the largest stretch of a unit vector. A linear map sends the unit sphere to an ellipsoid, possibly degenerate; its largest semiaxis has length

∥ A ∥_{2}

. Equivalently the matrix 2-norm is the largest singular value.

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