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Linear Algebra With Applications
Found in: Page 401
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Short Answer

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


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

p>q>-pn-1 where p,q0

See the step by step solution

Step by Step Solution

Step 1: Given Information:


Step 2: Finding the positive definite values:

Consider the following n×n matrix with the constants p and q:


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:


localid="1659688090121" p>qn-1q+p>0



As a result, we receive

p>q>-pn-1 where p,q0

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

Step 3: Determining the Result:

p>q>-pn-1 where p,q0

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