# Positive definite matrix

1. Feb 20, 2010

### 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

2. Feb 20, 2010

### Hurkyl

Staff Emeritus
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?

3. Feb 20, 2010

### 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

4. Feb 20, 2010

### Hurkyl

Staff Emeritus
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.

5. Feb 20, 2010

### Karthiksrao

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

Thanks

6. Feb 20, 2010

### Hurkyl

Staff Emeritus
I believe that statement.

7. Feb 22, 2010

### 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.