Positive definite quadratic forms proof

In summary, W. shows that if A is positive definite, then A = (B^T)B and that if A is positive semidefinite, then there exists a (possibly singular) real matrix Q such that A = (Q^T)Q. He also provides a solution to the second part of the homework equation if one can show that the assumption of D having all positive entries is necessary.
  • #1
wakko101
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0

Homework Statement


Given a real symmetric matrix A, prove that:

a) A is positive definite if and only if A = (B^T)B for some real invertible matrix B
b) A is positive semidefinite if and only if there exists a (possibly singular) real matrix Q such that A = (Q^T)Q

Homework Equations


quadratic form q(x) = a1*x1^2 + ... + an*xn^2
And possibly the principal axis theorem.

The Attempt at a Solution


for part a) I think I know how to show that A is pos def if you assume A = (B^T)B:
(P^T)AP = D = (P^T)(B^T)BP = ((PB)^T)PB which implies the diagonal entries of the diagonal matrix D are positive since row of (A^T) = column of A.
I'm not sure how to do it the other way, however. Any hints or advice would be appreciated.

Cheers,
W. =)
 
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  • #2
For the "only if" part of (a), do you know a theorem that allows you to express any symmetric matrix in the form

[tex]A = U^T D U[/tex]

where [itex]U[/itex] is a unitary matrix (in fact, orthogonal since [itex]A[/itex] is real in your case) and [itex]D[/itex] is a diagonal matrix? If [itex]A[/itex] is positive definite, what can you say about the elements of [itex]D[/itex]?

Also, I'm not sure I buy your proof of the "if" part. A matrix [itex]A[/itex] is positive definite if for every real vector [itex]x[/itex],

[tex]x^T A x \geq 0[/tex]

If [itex]A = B^T B[/itex], then what does that imply about [itex]x^T A x[/itex]?
 
  • #3
I'm not entirely sure this is correct, but I've come up with the following:

If (X^T)AX > 0 then we have (Y^T)DY > 0 (for X = PY and D = (P^T)AP). This implies that the entries on the diagonal matrix D (the eigenvalues) are all positive. So, write D = (Q^T)Q where Q is the diagonal matrix of the square roots of D's elements. Then
D = (Q^T)Q = (P^T)AP = (P^1)AP (since P is orthogonal)
then P(Q^T)Q(P^1) = A
then (PQ)(Q^T)(P^T) (since (Q^T) = Q)
then (PQ)(PQ)^T = A
Take B = (PQ)^T and we have the result.

As for the other part, I don't think I quite finished my reasoning before...if we can show that the assumption requires that D have all positive entries (take them to be a1 to an) then q(v) = a1*xa^2 + ... + an*xn^2 will necessarily be positive. And since D = (PB)^T(PB) then they must be positive because the diagonal entries of D (which are the only ones that are not 0) are the rows of the transpose of PB multiplied by the columns of PB and these are the same vectors.

Cheers,
W. =)
 

1. What is a positive definite quadratic form?

A positive definite quadratic form is a mathematical expression that involves the square of variables, with all coefficients being positive. It can be written in the form of Q(x) = xTAx, where x is a vector of variables and A is a symmetric matrix with positive eigenvalues.

2. How do you prove that a quadratic form is positive definite?

To prove that a quadratic form is positive definite, we need to show that for any non-zero vector x, Q(x) > 0. This can be done by using the eigenvalues of the matrix A and showing that all of them are positive. Another method is to use the Sylvester's criterion, which states that a symmetric matrix is positive definite if and only if all its leading principal minors are positive.

3. What is the importance of positive definite quadratic forms?

Positive definite quadratic forms have many applications in mathematics and other fields such as physics, engineering, and computer science. They are used to find the minimum or maximum values of a function, and to determine the nature of critical points. They also have important applications in optimization problems and in the study of differential equations.

4. Can a quadratic form be both positive definite and negative definite?

No, a quadratic form can only be either positive definite or negative definite. If a quadratic form is both positive definite and negative definite, it means that all its eigenvalues are both positive and negative, which is not possible. However, a quadratic form can also be indefinite, which means that it can take both positive and negative values depending on the input vector.

5. Are there any other criteria for positive definiteness of a quadratic form?

Yes, apart from the Sylvester's criterion, there are other criteria such as the Cholesky decomposition, which states that a symmetric positive definite matrix can be written as the product of a lower triangular matrix and its transpose. Another criterion is the positive definiteness of the Hessian matrix, which is the matrix of second derivatives of a multivariate function.

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