# Matrix Equation

#### Rodrigo_pn

##### New member
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

##### Full Member
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

##### New member
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

##### New member
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
Staff member
Response #2 gave you initial step - and asked you to finish!

Please show us how far you progressed with that hint.

#### Rodrigo_pn

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

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