Exercises

01

Eliminating cross terms.

Repeat the example to eliminate cross terms for the quadratic form $Q(\mathbf{x})=2x_1^2-4x_1{x}_2-x_2^2$.

02

Optimizing $Q(\mathbf{x})=2x_1^2-4x_1{x}_2-x_2^2$

Repeat the example on this quadratic form, using your work in Exercise 12-01, to find the min and max of $Q(\mathbf{x})=2x_1^2-4x_1{x}_2-x_2^2$ for $\mathbf{x}$ a unit vector, and find the unit vectors that produce these extreme values.

03

Quadratic form: eliminating cross terms and optimizing

• (a) Eliminate cross terms in the quadratic form $Q(\mathbf{x})=2x_1^2-4x_1{x}_2-x_2^2$.
• (b) Find the min and max values of $Q(\mathbf{x})$ from (a) where $\mathbf{x}$ is a unit vector, and find the unit vectors $\mathbf{x}$ that produce these extreme values. (Use your result in (a) to minimize in changed variables, and then change variables back again.)