Given two definite positive definite matrices(adsbygoogle = window.adsbygoogle || []).push({}); AandBof identical size with the following relationship of their diagonal elements:

[tex] A_{ii} \geq B_{ii}[/tex] (no summation)

which also holds after any unitary change of coordinates

[tex] \textbf{A}'=\textbf{U}^T\textbf{A}\textbf{U}[/tex]

where [tex]\textbf{U}[/tex] complete set of orthonormal vectors.

Question: What does this imply in terms of inequalities on the eigenvalues of the matrices?

The sum of eigenvalues ofAis of course equal of greater than the sum of eigenvaluesB:s eigenvalues, but this is true even without allowing for change of coordinates. I'm sure you must be able to deduce something stronger when the inequality holds under any unitary coordinate change...

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# Any implications of this diagonal element inequality?

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