Direct linear transformation: Difference between revisions

Revision as of 12:10, 3 August 2007

Revision as of 12:12, 3 August 2007

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where <math> [\mathbf{x}_{k}]_{\times} </math> is the [[Cross product#Conversion to matrix multiplication|matrix representation of the vector cross product]]. Notice that this last equation is a vector valued equation; the left hand side is the zero element in <math> \mathbb{R}^{3} </math>.

This means that eachEach value of ''k'' provides three homogeneous linear equations in the unknown elements of <math> \mathbf{A} </math>. However, since <math> [\mathbf{x}_{k}]_{\times} </math> has rank = 2, at most two equations are linearly independent. In practice, therefore, it is common to only use two of the three matrices <math> \mathbf{H}_{k} </math>, for example, for ''k''=1, 2. However, the linear dependency between the equations is dependent on <math> \mathbf{x}_{k} </math>, which means that in unlucky cases it would have been better to choose ''k''=2,3. As a consequence, if the number of equations is not a concern, it may be better to use all three equations when the matrix <math> \mathbf{B} </math> is constructed.

The linear dependence between the resulting homogeneous linear equations is a general concern for the case ''p > 2'' and has to be dealt with either by reducing the set of anti-symmeric matrices <math> \mathbf{H}_{k} </math> or by allowing <math> \mathbf{B} </math> to become larger than necessary for determining <math> \mathbf{a} </math>.