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Chapter 8: Q38E (page 401)

For which values of the constants p and q is the matrix

B=[pqLqqpLqMM0MqqLp]

positive definite? (B has p鈥檚 on the diagonal and q鈥檚 elsewhere.) Hint: Exercise 8.1.17 is helpful.

Short Answer

Expert verified

p>q>-pn-1wherep,q鈮�0

Step by step solution

01

Given Information:

B=pq鈰�qqp鈰�q鈰�鈰�鈰�鈰�qq鈰�p

02

Finding the positive definite values:

Consider the followingn脳nmatrix with the constants p and q:

B=pq鈰�qqp鈰�q鈰�鈰�鈰�鈰�qq鈰�p

The eigen values of the following matrix B are p-q (multiplicity n-1) and nq+p-q, respectively (multiplicity 1). Because all eigen values must be positive for matrix B to be positive definite, we have:

p-q>0nq+p-q>0

localid="1659688090121" 鈬�p>qn-1q+p>0

鈬�p>qn-1q>-p

鈬�p>qq>-pn-1

As a result, we receive

p>q>-pn-1wherep,q鈮�0

As a result, the matrix B is positive definite for these values of the constants p and q.

03

Determining the Result:

p>q>-pn-1wherep,q鈮�0

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(All ones in the last row and the last column, and zeros elsewhere.)
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