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I used to find it difficult to transform the Miller index in close-packed hexagonal (HCP). I mean, it should not be difficult theoretically, but it seems to be hard to understand. Thus, I spend some time to describe it in Matrix Form, in order to make it easy to grasp, and easy to calculate by programming.

Perhaps it can be the very first step for me to dig into more challenging problems (I feel sorry to say that I’m not a good English user, but I am trying to make it better).

OK, now let’s start.

# Inference

The 3-axis coordination $[UVW]$ represents the same direction as the 4-axis coordination $[uvtw]$. And, as we know, $[UVW]$ and $[uvtw]$ are both vectors, which can be described with their own primitive vectors. So we have:

In geometry, we have a relationship between vectors, this is:

Also, since it is satiated to describe a vector in a plane with two primitive vectors rather than three, we have relationship between indexes as below:

Having solved all the equations above, we get:

An easier description with matrix is:

In linear algebra, theoretically we can solve vector equation $Ab=x$ by method like $A^{−1}b$, namely, the inversed $A$ times the vector $b$. Therefore, the inversed description of the matrix formula is: