Positive definite matrix

In summary, the question is whether g'Bg is greater than or equal to g'g when B is a positive definite matrix and g is a vector. Upon further analysis, it is seen that this relation holds true for 1x1 matrices, as long as the eigenvalues of B are greater than 1. However, for larger matrices, it may not always hold true and additional variables need to be introduced for a more comprehensive proof.
  • #1
Karthiksrao
68
0
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
 
Physics news on Phys.org
  • #2
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
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
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
Yeah.. Understood. Eigen values of B has to be greater than 1 for the relation to hold true.

Thanks
 
  • #6
I believe that statement.
 
  • #7
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.
 

1. What is a positive definite matrix?

A positive definite matrix is a square matrix where all the eigenvalues are positive. This means that when we multiply the matrix by any non-zero vector, the resulting product is always positive.

2. What are the properties of a positive definite matrix?

Some properties of a positive definite matrix include: all eigenvalues are real and positive, all principal minors are positive, and it is nonsingular (meaning it has an inverse).

3. How is a positive definite matrix used in mathematics?

Positive definite matrices are used in many areas of mathematics, including optimization, statistics, and linear algebra. They are particularly useful in optimization problems, as they can help determine the minimum or maximum values of a function.

4. How do you determine if a matrix is positive definite?

To determine if a matrix is positive definite, we can use several methods. One method is to check the signs of the eigenvalues – if they are all positive, the matrix is positive definite. Another method is to check if all the principal minors are positive.

5. Can a matrix be both positive definite and positive semidefinite?

No, a matrix cannot be both positive definite and positive semidefinite. A positive semidefinite matrix has all non-negative eigenvalues, whereas a positive definite matrix has all positive eigenvalues. Therefore, a positive definite matrix is a stricter condition than positive semidefinite.

Similar threads

  • Linear and Abstract Algebra
Replies
1
Views
490
  • Linear and Abstract Algebra
Replies
12
Views
1K
  • Linear and Abstract Algebra
Replies
14
Views
1K
  • Linear and Abstract Algebra
Replies
8
Views
996
  • Linear and Abstract Algebra
Replies
6
Views
413
Replies
24
Views
1K
  • Linear and Abstract Algebra
Replies
20
Views
893
  • Linear and Abstract Algebra
Replies
2
Views
494
  • Linear and Abstract Algebra
Replies
1
Views
766
Replies
14
Views
1K
Back
Top