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I have come across the following equation but not sure how to solve it for matrix $D$. Let matrices $E$, $M$ and $D$ are three symmetric, $2\times 2$ and semi p.d. matrices. How to solve the equation $M = ED + DE$ for D, where $M$ and $E$ are known?

I tried the following, \begin{equation} M = ED + DE = E^{1/2}DE^{1/2}+E^{1/2}DE^{1/2} = E^{1/2}(D+D) E^{1/2} \end{equation} Then, we get $D = \frac{1}{2}E^{-1/2}ME^{-1/2}$. But I do not think it is correct as I found a counterexample, and also it shows $ED=DE$ which is not always true.

I appreciate any help in this regard.

S. Maths
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David
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1 Answers1

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This is called a sylvester equation, and there are many known ways to solve it (for instance, on the linked wikipedia article). Naively, the idea is to write $X$ as a vector rather than a matrix, and encode all the relevant linear conditions in one (bigger) matrix.

You can also find information at this wonderful blog post. Though now that you know the name, I'm confident you'll have no issues finding solution techniques!

I hope this helps ^_^

HallaSurvivor
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