Matrix Equation

Rodrigo_pn

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

Thank you
 

topsquark

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Aug 27, 2012
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379
Hello guys. I have a matrix equation: \(\displaystyle U^TPU\Delta + \Delta{U^TPU} = -I\), whose solution given is: \(\displaystyle P = (-1/2) (UAU^T) ^ {-1}\). The matrix \(\displaystyle P\) is hermitian and \(\displaystyle \Delta\) 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
 

Rodrigo_pn

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Joined
May 21, 2019
Messages
4
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
\(\displaystyle \Delta\)
 

Rodrigo_pn

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Joined
May 21, 2019
Messages
4
Hello guys. I have a matrix equation: \(\displaystyle U^TPU\Delta + \Delta{U^TPU} = -I\), whose solution given is: \(\displaystyle P = (-1/2) (U\Delta{U^T) ^ {-1}}\). The matrix \(\displaystyle P\) is hermitian and \(\displaystyle \Delta\) is a diagonal matrix. How to reach this solution?

Thank you
 

Subhotosh Khan

Super Moderator
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Jun 18, 2007
Messages
18,353
Response #2 gave you initial step - and asked you to finish!

Please show us how far you progressed with that hint.
 

Rodrigo_pn

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Joined
May 21, 2019
Messages
4
Response #2 gave you initial step - and asked you to finish!

Please show us how far you progressed with that hint.
The continuation: \(\displaystyle U^TPU\Delta+\Delta{U^TPU}=-I\)
\(\displaystyle U^TPU\Delta+U^TPU\Delta = -I\)
\(\displaystyle U^{-T}[2U^TPU\Delta=-I]U^{-1}\Delta^{-1}\)
\(\displaystyle P=-\dfrac{1}{2}U^{-T}U^{-1}\Delta^{-1}\)
\(\displaystyle P=-\dfrac{1}{2}U^{-T}\Delta^{-1}{U^{-1}}\)
\(\displaystyle P=-\dfrac{1}{2}(U\Delta{U^T})^{-1}\)
 
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