Positive definite symmetric matrix has LDL^T decomposition with positive diagonals

QUESTION:

Let be an matrix with a factorization , where is a unit lower triangular matrix and is a diagonal matrix. Show that is positive definite if and only if , for .

ATTEMPT:

Assume for . Then is possible and

where .

Because has pivots it is nonsingular and hence if and only if . Therefore for all .

Assume is positive definite. How can I prove that has positive diagonals?

Re: Positive definite symmetric matrix has LDL^T decomposition with positive diagonal

Hey math2011.

Isn't one of the definitions of a positive define matrix having positive eigenvalues?

Positive-definite matrix - Wikipedia, the free encyclopedia

Re: Positive definite symmetric matrix has LDL^T decomposition with positive diagonal

Quote:

Originally Posted by

**chiro**

Yes, but how can I show that the diagonal entries of are the eigenvalues of ?

Re: Positive definite symmetric matrix has LDL^T decomposition with positive diagonal

Is your L matrix the matrix of eigenvectors and also is that matrix orthonormal? If so then L^T = L^(-1) and you have a eigen-decomposition of your matrix G which means that D is your matrix of eigenvalues.

Re: Positive definite symmetric matrix has LDL^T decomposition with positive diagonal

I don't know if L is the matrix of eigenvectors. The problem does not state that explicitly. Are you suggesting that L actually is the matrix of eigenvectors? How can I prove it?

Re: Positive definite symmetric matrix has LDL^T decomposition with positive diagonal

My suggestion is to show that LDL^t is the diagonal form by showing that L is non-singular (since it is triangular with non-zero determinant) and then use this to show that L is also orthonormal.

For the ortho-normal aspect, if you can show L*L^t = I then you're done.

By showing that L^t = L^(-1) and by using the assumption that the diagonalization produces a unique decomposition, then you have shown that this is indeed the diagonalization.

Re: Positive definite symmetric matrix has LDL^T decomposition with positive diagonal

Thanks for explaining but I still cannot see how to do it. L^t is a unit upper triangular matrix and L is a unit lower triangular matrix, L^t L does not necessarily equal to the identity matrix unless the L is special. The problem does not state any other information about L that directly say L is orthogonal.