Show that if f is differentiate at a with Jacobian matrix Jf(a), then Here ||Jf(a)|| is th...

Show that if f is differentiate at a with Jacobian matrix Jf(a), then

Here ||Jf(a)|| is the matrix norm defined in Exercise 1. [Hint: Write f(x) − f(a) = (f(x) − f(a) − Jf(a)(x − a)) + Jf(a)(x − a), use the triangle inequality, use the differentiability of f at a, and use Exercise 2.]

Exercise 1–2

1. The norm of a m × n matrix A = [aij] is

Calculate the norm of the following matrices.

(a)

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(b)

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(c)

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(d)

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(e)

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(f)

2. Prove that with the norm of a matrix defined as in Exercise 1,

 ∥Ax∥ ≤ ∥A∥∥x∥ for any vector x.

[Hint: By (2.2.2), Ax = ∑ xiai, where ai is the ith column of A. Use the triangle inequality and the Cauchy-Schwarz inequality.]