Q4E If the matrix [abbc] is positive... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 8: Q4E (page 413)

If the matrix $[\begin{matrix} a & b \\ b & c \end{matrix}]$ is positive definite, then $' a'$ must be positive.

Short Answer

Expert verified

The given statement is TRUE.

Step by step solution

Definition of a positive definite matrix:

A symmetric matrix is positive definite if and only if all of its Eigen values are positive. Also, the matrix is positive semi definite if and only if all of its Eigen values are positive or zero.

To Find TRUE or FALSE

The given matrix is

$A = [\begin{matrix} a & b \\ b & c \end{matrix}]$

If the given matrix is positive definite, this means both Eigen values must be positive.

This means $q (\overset{鈫�}{x}) = \overset{鈫�}{x} 路 A \overset{鈫�}{x} > 0$ for all non-zero $\overset{鈫�}{x}$ .

Thus, compute $0 < \overset{鈫�}{x} 路 A \overset{鈫�}{x} 鈫�$ as follows:

$0 < \overset{鈫�}{x} 路 A \overset{鈫�}{x} 鈫� 0 < [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] [\begin{matrix} a & b \\ b & x \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] 鈫� 0 < [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] [\begin{matrix} a x_{1} + b x_{2} \\ b x_{1} + c x_{2} \end{matrix}] 鈫� 0 < a {x_{1}}^{2} + 2 b x_{1} x_{2} + c {x_{2}}^{2} 鈫�$

From above, it is clear that must be positive such that this claim holds for all non-zero

\overset{鈫�}{x}

Thus, the given statement is TRUE.

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91影视

If the matrix $[\begin{matrix} a & b \\ b & c \end{matrix}]$ is positive definite, then $' a'$ must be positive.

Short Answer

Step by step solution

Definition of a positive definite matrix:

To Find TRUE or FALSE

One App. One Place for Learning.

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91影视

If the matrix [abbc] is positive definite, then'a'must be positive.

Short Answer

Step by step solution

Definition of a positive definite matrix:

To Find TRUE or FALSE

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Decision Maths

Theoretical and Mathematical Physics

Geometry

Pure Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

If the matrix $[\begin{matrix} a & b \\ b & c \end{matrix}]$ is positive definite, then $' a'$ must be positive.