Describe the norms that are generated by the inner products presented in Example 5.3.1.
• Given a nonsingular matrix A ∈ \mathcal{C}^{n×n}, the A-norm (or elliptical norm) generated by the A-inner product on \mathcal{C}^{n×1} is
||x||_A = \sqrt{〈x|x〉} = \sqrt{x^∗A^∗Ax} = ||Ax||_2. (5.3.5)
• The standard inner product for matrices generates the Frobenius matrix norm because
||A|| = \sqrt{〈x|x〉} = \sqrt{trace (A^∗A)} = ||A||_F. (5.3.6)
• For the space of real-valued continuous functions defined on (a, b), the norm of a function f generated by the inner product 〈f|g〉 = \int_{a}^{b} f(t)g(t)dt is
||f|| = \sqrt{〈f|f〉} = \left(\int_{a}^{b}{f(t)^2 dt}\right)^{1/2}.