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Chapter 8: Q9E (page 413)

If matrix Ais positive definite, then all the eigenvalues ofmust be positive.

Short Answer

Expert verified

The given statement is TRUE.

Step by step solution

01

Step 1: Definition of positive definite matrix:

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

02

Step 2: To Find TRUE or FALSE

ConsiderA=1-1-13

To prove that the given matrix is positive definite, find the symmetric of the matrix.

Because if the matrix is a symmetric matrix, then it is said to be a positive definite matrix.

AT=1-1-13=A

Here, AT=Awhich means it鈥檚 symmetric. So, the given matrix is positive definite.

Find the eigenvalues:

(A-位I)=0111-1-位1001=0111-1-位00位=0

Simplify further,

1-位11-1-位=01-位-1-位-11=01-位-位+位2-1=0位2-2位=0位位-2=0位=0,2

The matrix A has positive eigenvalues, and it鈥檚 a positive definite matrix.

Hence, the statement isTRUE.

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Most popular questions from this chapter

If Ais an invertible symmetric matrix, thenA2must be positive definite.

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