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Chapter 1: Q25E (page 1)

If the positive definite matrix Ais similar to the symmetric matrix B, then Bmust be positive definite as well.

Short Answer

Expert verified

The given statement is TRUE.

Step by step solution

01

Check whether the given statement is TRUE or FALSE

It is given that A is a positive definite matrix. So,A is similar to a diagonal matrix whose all diagonal entries are positive.

Also,B is similar matrix to A.

So, there exists a non-singular matrix C such that BC=A.

As a result,BC is similar to a diagonal matrix whose all diagonal entries are positive.

Since C is non-singular, therefore is similar to a diagonal matrix whose all diagonal entries are positive.

Thus,B is also positive definite.

02

Final Answer

The given statement is TRUE.

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