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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 ?
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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.

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  • $\begingroup$ 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 ? $\endgroup$ – nurabha Aug 25 '15 at 7:55
  • $\begingroup$ Yes, take the eigenvalues of a positive definite matrix $\endgroup$ – Omnomnomnom Aug 25 '15 at 11:01
  • $\begingroup$ Ok, so only if A is positive definite itself that its Eigenvalue matrix will be positive definite. Makes some sense $\endgroup$ – nurabha Aug 25 '15 at 13:52
  • $\begingroup$ @nurabha it's obvious when you realize that the eigenvalues of a diagonal matrix are just its diagonal entries. $\endgroup$ – Omnomnomnom Aug 25 '15 at 13:53
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    $\begingroup$ 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! $\endgroup$ – nurabha Aug 25 '15 at 14:27

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