On Aug 24, 2006, at 8:02 PM, Andy Little wrote:
"Matthias Troyer" <troyer@itp.phys.ethz.ch> wrote
Andy Little wrote <...>
For matrices my tests have been limited to homogeneous transformation matrices. In this case the matrix elements for a *simple/obvious* implementation of a matrix for a physical quantity are comprised of various types, comprising of the quantity type, the reciprocal of the quantity and numeric types. It may be possible to somehow 'tag' a numeric matrix as a quantity matrix, but I havent invetigated that.
Just use a quantity as value type? What's the problem with that?
Assume a 2D homogeneous transform matrix intended to transform 2d points. With 9 elements it is possible for the matrix to contain up to 9 separate types T0 - T9
( T1 , T2, T3) ( T4, T5, T6 ) ( T7, T8, T9)
In the transform matrix T1, T2, T4 and T5 are concerned with rotation. IOW to rotate a point through an angle alpha
(cos(alpha), sin(alpha), T3 (-sin(alpha), cos(alpha), T6) ( T7, T8, T9)
Therefore T1, T2, T4, T5 must be numeric.
T7 and T8 represent translation. To translate a homogeneous point [ X Y h], assuming h is a numeric and X and Y are some quantity, T7 and T8 must have the dimensionally equivalent quantity type ( but not necesarily same unit) as X and Y. (In fact in this case T3 and T6 have the reciprocal quantity type of X and Y and T9 is always a numeric representing scaling)
Wait a moment, matrices are representations of linear operators in a vector space Let us first consider what that vector space here is. For the transformation matrices you mention we have vectors (x, y, 1) that we want to transform. We can choose, as is usual, the same units for the three components. If we now have a linear operator from this vector space to itself, as we have here, then all entries in the matrix must be pure numbers, since we do not change any units. And indeed, if one looks at the matrix you propose than for translations + rotations it actually is (cos(alpha), sin(alpha), dx ) (-sin(alpha), cos(alpha), dy) (0, 0, 1) all entries are pure numbers, and the translations dx and dy are given in terms of the unit chosen for the 1 in the vector. This is actually just what you propose as second option:
Another alternative is to make all the elements of the matrix numeric. In this case h in the homogeneous coordinate [X Y h] would have the same type as X and Y.
But I disagree with your next statement:
That isnt as neat though as you have then lost the unit information in T7 and T8 regarding the translation and applying the matrix to a point in different units would give different results.
No, you have not lost the unit information, since it is encoded in the vector type. And indeed applying a matrix to different vector spaces (e.g. points with different units), should actually give different results!
I hope I have demonstrated some of the issues in the above example. However I have only limited my experiments to transform matrices. I should probably look at other matrix calculations to see what happens in other cases too.
From my 20 years of experience with matrix calculations I have not seen a case yet where different value types are required for different matrix elements - this actually contradicts the definition f a matrix, where all elements are taken from the same number field. Matthias