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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 8: Q41E (page 414)

If matrix $[\begin{matrix} a & b & c \\ b & d & e \\ c & e & f \end{matrix}]$ is positive definite, then $af$ must exceed $c^{2}$ .

Short Answer

Expert verified

The given statement is FALSE.

Step by step solution

01

Step 1: Definition of positive definite matrix

A matrix is said to be positive definite matrix, when it is symmetric matrix and all its eigenvalues are positive.

02

To Find TRUE or FALSE

It is given that $A = [\begin{matrix} a & b & c \\ b & d & e \\ c & e & f \end{matrix}]$ is a positive definite matrix.

So, the principal sub matrices are $A^{(0)} = [a], A^{(2)} = [\begin{matrix} a & b \\ b & d \end{matrix}]$ and $A^{(3)} = [\begin{matrix} a & b & c \\ b & d & e \\ c & e & f \end{matrix}]$ such that

$\det (A^{(1)}) = a > 0 鈰� (1) \det (A^{(2)}) = ad - b^{2} 鈰� (2) \det (A^{(3)}) = adf - 2 bce - {ae}^{2} - b^{2} f - c^{2} d > 0 鈰� (3)$

03

Final Answer

The 3rd equation can be written as

$(ad - b^{2}) f > {ae}^{2} + c^{2} d - 2 bce$

Using equations 1 and 2, we are left with

$(ad - b^{2}) f > c^{2} d - 2 bce while {ae}^{2} > 0$

From this, it is not confirm that $af - c^{2} > 0 or af > c^{2}$ .

Thus, the given statement is FALSE.

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Most popular questions from this chapter

Show that the diagonal elements of a positive definite matrix A are positive.

Consider the quadratic form

$q (x_{1}, x_{2}) = {ax}_{1}^{2} + {bx}_{1} x_{2} + {cx}_{2}^{2}$ .

We define

$q_{11} = \frac{{鈭�}^{2} q}{鈭� x_{1}^{2}}, q_{12} = q_{21} = \frac{{鈭�}^{2} q}{鈭� x_{1} 鈭� x_{2},}, q_{22} = \frac{{鈭�}^{2} q}{鈭� x_{2}^{2}}$ .

The discriminant D of q is defined as

$D = \det [\begin{matrix} q_{11} & q_{12} \\ q_{21} & q_{22} \end{matrix}] = q_{11} q_{22} - {(q_{22})}^{2}$ .

The second derivative test tells us that if D androle="math" localid="1659684555469" $q_{11}$ are both positive, then

$q (x_{1}, x_{2})$ has a minimum at (0, 0). Justify this fact, using the theory developed in this section.

For which angle(s) $胃$ can you find four distinct unit vectors in ${鈩�}^{3}$ such that the angle between any two of them is $胃$ ? Draw a sketch.

54. If Aand B are real symmetric matrices such that, $A^{3} = B^{3}$ thenmust be equal to B.

Let A be a $2 脳 2$ matrix and $\overset{鈫�}{u}$ a unit vector in ${鈩�}^{2}$ . Show that $蟽_{2} 猢� A \overset{鈫�}{u} 猢� 蟽_{1},$

where $蟽_{1}, 蟽_{2}$ are the singular values of A. Illustrate this inequality with a sketch, and justify it algebraically.

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