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The Frame of Reference Transformation Matrix (3006,00C6) ^AM_B describes how to transform a point (^Bx,^By,^Bz) with respect to RCS_B into (^Ax,^Ay,^Az) with respect to RCS_A according to Equation C.20.2-1.

The Matrix Registration is expressible as multiple matrices, each in a separate Item of Matrix Sequence (0070,030A). Equation C.20.2-2 specifies the order of the matrix multiplication where M₁ , M₂ and M₃ are the first, second and third Items in the Sequence.

where

Registration often involves two or more RCS, each with a corresponding Frame of Reference Transformation Matrix. For example, another Frame of Reference Transformation Matrix ^AM_C can describe how to transform a point ( ^Cx,^Cy,^Cz) with respect to RCS _C into ( ^Ax,^Ay,^Az) with respect to RCS_A. It is straightforward to find the Frame of Reference Transformation Matrix ^B M _C that describes how to transform the point ( ^Cx,^Cy,^Cz) with respect to RCS_C into the point ( ^Bx,^By,^Bz) with respect to RCS_B. The solution is to invert ^AM_B and multiply by ^AM_C, as shown in Equation C.20.2-3.

If two or more transformation matrices describe the relation between Patient coordinates and a device-centric Well-known Frame of Reference, any calculations assuming transitivity via the Well-known Frame of Reference must be performed with great care to assure that both registrations reflect the same positioning of the patient with respect to the common Well-known Frame of Reference.