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Chapter 8: Q8.2-28E (page 401)

Show that any positive definite n x n matrix A can be written as A=BBT, where B is a n x n matrix with orthogonal columns. Hint: There exists an orthogonal matrix S such that S-1AS = STAS = D is a diagonal matrix with positive diagonal entries. Then A=SDST. Now write D as the square of a diagonal matrix.

Short Answer

Expert verified

It has been proved that A can be written as A=BBT.

Step by step solution

01

Results used

Given that A is a n x n positive definite matrix.

It is known that there exists an orthogonal matrix S such that S-1AS=STAS=Dwhere D is a diagonal matrix with positive diagonal entries.

Then it is written as A=SDST.

Now write D as the square of a diagonal matrix dwhere entries of d are equal to square root of that of D.

That is D = d2.

02

Write A = BBT

Since, it is written as:

A=SDST=Sd2ST=SddTST=SdSdT

Now, let Sd = B.

So, it is written as A = BBT.

Hence, A can be written as A = BBT.

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