Page 717 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Express the quadratic form in the matrix notation , where A is symmetric.
15.

Let . In statistics, the quantity
16.

is called the sample mean of , , …, , and

is called the sample variance.

(a) Express the quadratic form in the matrix notation , where A is symmetric.
(b) Is a positive definite quadratic form? Explain.

Complete the proof of Theorem 9.5.1 by showing that if and if x is an eigenvector of A
17. corresponding to .

Indicate whether each statement is true (T) or false (F). Justify your answer.
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(a) A symmetric matrix with positive entries is positive definite.

(b) is a quadratic form.

(c) is a quadratic form.

(d) A positive definite matrix is invertible.

(e) A symmetric matrix is positive definite, negative definite, or indefinite.

(f) If A is positive definite, then is negative definite.

Indicate whether each statement is true (T) or false (F). Justify your answer.
19.

(a) If x is a vector in , then is a quadratic form.