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There are some methods mentioned on Wikipedia. I would select whichever of these is least complex, but still efficient for the size of the matrix you are working with. Once the algorithm is selected, you need to map into hardware. I would probably implement as a systolic array or data-flow processor, as it is easy to use them to map mathematical equations into hardware. There are many papers on how to do this.
What sort of things have you tried? Why were they unsuitable?
for example, if you use QL or QR algorithm to slove eigenvalue, you cannot determine the loop count before the loop finish, while it is bad for synthesis, and you cannot know how many cycles it will cost,
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