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Q38E

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Found in: Page 401

### Linear Algebra With Applications

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

# For which values of the constants p and q is the matrix ${\mathbf{B}}{\mathbf{=}}\left[\begin{array}{cccc}p& q& L& q\\ q& p& L& q\\ M& M& 0& M\\ q& q& L& p\end{array}\right]$ positive definite? (B has p’s on the diagonal and q’s elsewhere.) Hint: Exercise 8.1.17 is helpful.

$\mathrm{p}>\mathrm{q}>-\frac{\mathrm{p}}{\mathrm{n}-1}\mathrm{where}\mathrm{p},\mathrm{q}\ne 0$

See the step by step solution

## Step 1: Given Information:

$\mathrm{B}=\left[\begin{array}{cccc}\mathrm{p}& \mathrm{q}& \cdots & \mathrm{q}\\ \mathrm{q}& \mathrm{p}& \cdots & \mathrm{q}\\ ⋮& ⋮& \ddots & ⋮\\ \mathrm{q}& \mathrm{q}& \cdots & \mathrm{p}\end{array}\right]$

## Step 2: Finding the positive definite values:

Consider the following $\mathrm{n}×\mathrm{n}$ matrix with the constants p and q:

$\mathrm{B}=\left[\begin{array}{cccc}\mathrm{p}& \mathrm{q}& \cdots & \mathrm{q}\\ \mathrm{q}& \mathrm{p}& \cdots & \mathrm{q}\\ ⋮& ⋮& \ddots & ⋮\\ \mathrm{q}& \mathrm{q}& \cdots & \mathrm{p}\end{array}\right]$

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:

$\left\{\begin{array}{l}\mathrm{p}-\mathrm{q}>0\\ \mathrm{nq}+\mathrm{p}-\mathrm{q}>0\end{array}\right\$

localid="1659688090121" $⇒\left\{\begin{array}{l}\mathrm{p}>\mathrm{q}\\ \left(\mathrm{n}-1\right)\mathrm{q}+\mathrm{p}>0\end{array}\right\$

$⇒\left\{\begin{array}{l}\mathrm{p}>\mathrm{q}\\ \left(\mathrm{n}-1\right)\mathrm{q}>-\mathrm{p}\end{array}\right\$

$⇒\left\{\begin{array}{l}\mathrm{p}>\mathrm{q}\\ \mathrm{q}>-\frac{\mathrm{p}}{\left(\mathrm{n}-1\right)}\end{array}\right\$

$\mathrm{p}>\mathrm{q}>-\frac{\mathrm{p}}{\mathrm{n}-1}\left(\mathrm{where}\mathrm{p},\mathrm{q}\ne 0\right)$

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

## Step 3: Determining the Result:

$\mathrm{p}>\mathrm{q}>-\frac{\mathrm{p}}{\mathrm{n}-1}\mathrm{where}\mathrm{p},\mathrm{q}\ne 0$