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The wikipedia page for Cholesky decomposition says:

For real matrices, the factorization has the form $A = LDL^T$ and is often referred to as LDLT decomposition. It is closely related to the eigendecomposition of real symmetric matrices, $A = QΛQ^T$.

How are they closely related?

More specifically, what is the relation between the values on the diagonal of the LDLT decomposition and the eigenvalues on the diagonal of the eigendecomposition? (other than their similar signatures guaranteed by Sylvester's law of inertia)

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    $\begingroup$ math.stackexchange.com/questions/2465769/… $\endgroup$
    – Will Jagy
    Commented Oct 10, 2017 at 17:38
  • $\begingroup$ @WillJagy Deleted from there since it seems to a better fit here according to the declared purposes of both sites. $\endgroup$
    – Danra
    Commented Oct 10, 2017 at 19:46

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I don't think there's much more than what you already said on the signs and the fact that their products must be the same (comparing determinants).

In particular, the $LDL^T$ decomposition can be computed within the same base field as the coefficients (for instance, a matrix with rational entries has a rational $LDL^T$ decomposition), while an eigendecomposition often requires field extensions. This is, in my view, the most convincing argument as to why the two are fundamentally different objects.

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  • $\begingroup$ though, in the case of real symmetric matrices, the eigenvalues are also guarnateed to be real. $\endgroup$
    – Danra
    Commented Oct 10, 2017 at 21:55
  • $\begingroup$ Federico, if you use other stackexchange sites, you can edit those profiles to suit those sites. There should be a choice, at the point of saving your edits, of either exporting to all sites or just using the edits on that site. Meanwhile, on the matrix question, I asked about references here: math.stackexchange.com/questions/1388421/… $\endgroup$
    – Will Jagy
    Commented Oct 11, 2017 at 0:36

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