Jacobian Eigenvalue Algorithm and Positive definiteness of Eigenvalue matrix

Question

For a real symmetric matrix A of size n x n, the Jacobian Eigenvalue Algorithm produces

• n - Eigen values of A in the form of a Square Diagonal Eigenvalue Matrix of order n
• n - Eigen vectors of A corresponding to each of the n-Eigen values of A in the form of a n x n Matrix with columns as Eigen vectors.

Also, a matrix is said to be positive definite if it’s symmetric and all its eigenvalues are non-negative i.e. every Eigenvalue > 0.

Does the Jacobi Eigenvalue algorithm guarantees producing a positive definite Eigenvalue diagonal matrix for every real symmetric matrix A ?

If the answer to above queestion is no, then

• What must be changed so that Jacobi algorithm guarantees producing a positive definite Eigenvalue diagonal matrix ?
• If Jacobi Algorithm cannot guarantee that, are there any alternative iterative methods that produce n-Eigenvalues and n-Eigen vectors but guarantee producing a positive definite Eigenvalue matrix ?

Answer 1

The eigenvalue algorithm produces a diagonal matrix containing the eigenvalues of $A$. That matrix is positive definite if and only if the eigenvalues of $A$ are all positive.

So, generally we will not get a positive definite eigenvalue matrix.