# Jacobian Eigenvalue Algorithm and Positive definiteness of Eigenvalue matrix

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 ?

## 1 Answer

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.

• Thanks for answering. 1st part of your answer is same as I stated in the question. About the second part of your answer, is their a way to always get a positive definite eigenvalue matrix ? – nurabha Aug 25 '15 at 7:55
• Yes, take the eigenvalues of a positive definite matrix – Omnomnomnom Aug 25 '15 at 11:01
• Ok, so only if A is positive definite itself that its Eigenvalue matrix will be positive definite. Makes some sense – nurabha Aug 25 '15 at 13:52
• @nurabha it's obvious when you realize that the eigenvalues of a diagonal matrix are just its diagonal entries. – Omnomnomnom Aug 25 '15 at 13:53
• I am new to Diagnolization based on iterative methods, so probably didn't get the obvious concepts initially. Now I get them. Thanks for help! – nurabha Aug 25 '15 at 14:27