Chapter 7.2, Problem 22E

In Exercises 21 and 22, matrices are n × n and vectors are in ℝⁿ. Mark each statement True or False Justify each answer.

22.

1. a. The expression ||x||² is not a quadratic form.
2. b. If A is symmetric and P is an orthogonal matrix, then the change of variable x = Py transforms x^TAx into a quadratic form with no cross-product term.
3. c. If A is a 2 × 2 symmetric matrix, then the set of x such that x^TAx = c (for a constant c) corresponds to either a circle, an ellipse, or a hyperbola.
4. d. An indefinite quadratic form is neither positive semidefinite nor negative semidefinite.
5. e. If A is symmetric and the quadratic form x^TAx has only negative values for x ≠ 0, then the eigenvalues of A are all positive.