Chapter 7.2, Problem 21E

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

21.

1. a. The matrix of a quadratic form is a symmetric matrix.
2. b. A quadratic form has no cross-product terms if and only if the matrix of the quadratic form is a diagonal matrix.
3. c. The principal axes of a quadratic form x^TAx are eigenvectors of A.
4. d. A positive definite quadratic form Q satisfies Q(x) > 0 for all x in ℝⁿ.
5. e. If the eigenvalues of a symmetric matrix A are all positive, then the quadratic form x^TAx is positive definite.
6. f. A Cholesky factorization of a symmetric matrix A has the form A = R^TR, for an upper triangular matrix R with positive diagonal entries.