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Chapter 8: Q36E (page 414)

If Aand Bare positive definitematrices, then matrix A+Bmust be positive definite as well.

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 positive definite matrix, when it is symmetric matrix and all its eigenvalues are positive.

02

Step 2: To Find TRUE or FALSE

Since A and B are positive definite matrices, therefore xTAx>0 and xTBx>0 for all vectors x.

This implies xT(A+B)x = xTAx + xTBx > 0 for all x, which means that A+B is a positive definite.

For example; consider and

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

Here, AT= A and BT= Bwhich means it鈥檚 symmetric. So, the given matrix is positive definite.

Hence, the given statement is TRUE.

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