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I am taking differential equations in college right now and lately we have been working on eigen values and vectors. Beyond all the math I'm trying to conceptualize what is going on when you attain a solution vector. I understand that the eigen values represents the roots of the characteristic polynomial. What do eigen vectors physically represent or how do they affect the calculation of a physical system. I'm just trying to get a better understanding behind what they represent. I've looked elsewhere but can't seem to find an answer beyond a formula on how to find them.

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