Matrix Transformations in Crystallography

point group

1

matrix size

3×3

3×6

6×3

6×6

symmetry

no

yes

Original

m 11	m 12	m 13
m 21	m 22	m 23
m 31	m 32	m 33

1

Identity

1	0	0
0	1	0
0	0	1

TotalTotal

m

11

m

12

m

13

m

21

m

22

m

23

m

31

m

32

m

33



Final result

This Demonstration shows how various symmetry operations are used to simplify matrices; this process is called the direct inspection method. Select a crystallographic point group, the dimension of the matrix representation (number of rows × number of columns), and whether the original matrix should be assumed symmetric (only square matrices can be symmetric). The top row is the original untransformed matrix and the next row down displays the symmetry operations needed for the selected point group with the associated transformation matrix. The bottom row shows the result of each successive transformation step from left to right. It is in the last row that the matrix is progressively simplified, thus reducing the number of independent matrix elements.