Matrix Equation

Rodrigo_pn

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Joined
May 21, 2019
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Hello guys. I have a matrix equation: [MATH]U^TPU\Delta + \Delta{U^TPU} = -I[/MATH], whose solution given is: [MATH]P = (-1/2) (UAU^T) ^ {-1}[/MATH]. The matrix [MATH]P[/MATH] is hermitian and [MATH]\Delta[/MATH] is a diagonal matrix. How to reach this solution?

Thank you
 
Hello guys. I have a matrix equation: [MATH]U^TPU\Delta + \Delta{U^TPU} = -I[/MATH], whose solution given is: [MATH]P = (-1/2) (UAU^T) ^ {-1}[/MATH]. The matrix [MATH]P[/MATH] is hermitian and [MATH]\Delta[/MATH] is a diagonal matrix. How to reach this solution?

Thank you
Where did the A come from? It's just \(\displaystyle \Delta\). A typo maybe?

Anyway, a diagonal matrix commutes with anything so
\(\displaystyle U^TPU \Delta + \Delta U^TPU = U^TPU \Delta + U^TPU \Delta = \left ( U^TPU + U^TPU \right ) \Delta = -I\)

Can you finish?

-Dan
 
Where did the A come from? It's just \(\displaystyle \Delta\). A typo maybe?

Anyway, a diagonal matrix commutes with anything so
\(\displaystyle U^TPU \Delta + \Delta U^TPU = U^TPU \Delta + U^TPU \Delta = \left ( U^TPU + U^TPU \right ) \Delta = -I\)

Can you finish?

-Dan

Yes, it is a typo.
In place of matrix A is
[MATH]\Delta[/MATH]
 
Hello guys. I have a matrix equation: [MATH]U^TPU\Delta + \Delta{U^TPU} = -I[/MATH], whose solution given is: [MATH]P = (-1/2) (U\Delta{U^T) ^ {-1}}[/MATH]. The matrix [MATH]P[/MATH] is hermitian and [MATH]\Delta[/MATH] is a diagonal matrix. How to reach this solution?

Thank you
 
Response #2 gave you initial step - and asked you to finish!

Please show us how far you progressed with that hint.
 
Response #2 gave you initial step - and asked you to finish!

Please show us how far you progressed with that hint.

The continuation: [MATH]U^TPU\Delta+\Delta{U^TPU}=-I[/MATH][MATH]U^TPU\Delta+U^TPU\Delta = -I[/MATH][MATH]U^{-T}[2U^TPU\Delta=-I]U^{-1}\Delta^{-1}[/MATH][MATH]P=-\dfrac{1}{2}U^{-T}U^{-1}\Delta^{-1}[/MATH][MATH]P=-\dfrac{1}{2}U^{-T}\Delta^{-1}{U^{-1}}[/MATH][MATH]P=-\dfrac{1}{2}(U\Delta{U^T})^{-1}[/MATH]
 
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