 Solution

ID: past-exam-of-the-numerical-linear-algebra-course-of-the-university-of-oxford/2011/2/b/solution

y = A^{1/2} x

gives

∥ B ∥_{A} = sup_{x \neq = 0} \frac{∥ B x ∥ _{A}}{∥ x ∥ _{A}} = sup_{y \neq = 0} \frac{∥ A ^{1/2} B A ^{- 1/2} y ∥ _{2}}{∥ y ∥ _{2}} = ∥ A^{1/2} B A^{- 1/2} ∥_{2} .

The last expression is the Euclidean operator norm. This is the matrix norm induced by an energy norm: transforming to coordinates with Euclidean lengths transforms the operator by the corresponding matrix similarity.

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