Does g'Bg Compare with g'g When B is a Positive Definite Matrix?

[Karthiksrao]

I am curious to know how g'Bg compares with g'g when B is a positive definite matrix and g is a vector.

Is g'Bg >= g'g ?


Thanks,
Karthik

 

[Hurkyl]

Well, let's look at the simplest case.

No wait, 0x0 matrices are too simple. Let's try the next simplest case -- 1x1 matrices.

Try analyzing the 1x1 case. What do you see?

 

[Karthiksrao]

The relation seems to hold true in this case. Say, if B is 3, then g' *3*g is definitely greater than g'g

 

[Hurkyl]

Well, there are more 1x1 matrices than just [3]! You should try a few.

Anyways, rather than looking at 1x1 matrices one at a time, you should try proving it for all 1x1 matrices at once! You'll need to introduce one or more additional variables, of course.

 

[Karthiksrao]

Yeah.. Understood. Eigen values of B has to be greater than 1 for the relation to hold true.

Thanks

 

[Hurkyl]

I believe that statement.

 

[jvc]

I think it depends on the eigenvalues of B
if the eigenvalues of B is larger than 1, the statement holds. Otherwise, it does not.