Transforming a 6x6 stiffness matrix

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Hello...
I have a 6x6 stiffness matrix. I also have a 3x3 matrix containing the direction cosines of the x',y', and z' reference system. My question is: How can I transform my 6x6 stiffness matrix or how can I get a 6x6 transformation matrix?
 
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Do you mean if given the transformation x' = Ax, where A is a 6x6 matrix that how do we get another transformation of the form y' = By?
If so, we assume that x relates y with a given matrix operator say C (x = Cy). (dont mind my notation). We want to get the matrix B. How do you do this? Clearly x' = Cy'. N.B. C is assumed to be a non singular matrix. By substitution, we have y' = C^-1ACy, Thus by comparison we have B = C^-1AC. This implies the 6x6 matrix you obtained must be similar to the matrix A defined initially. B is the required matrix
 

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