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m rm fancy boxes around equations per style agreement reached a while ago in the math wikproject. Proper punctuation. |
→Example ''p''=3: Proper spacing in this expression. Also, the digit "2" should not be within the italics. See Wikipedia:Manual of Style (mathematics). |
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where <math> \mathbf{H}_{m} </math> is a ''M''-dimensional basis of the space of <math> p \times p </math> anti-symmetric matrices.
=== Example ''p'' = 3 ===
In the case that ''p'' = 3 the following three matrices <math> \mathbf{H}_{m} </math> can be chosen
: <math> \mathbf{H}_{1} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix} </math>, <math> \mathbf{H}_{2} = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{pmatrix} </math>, <math> \mathbf{H}_{3} = \begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} .</math>
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Each 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}_{m} </math>, for example, for ''m''=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, for example, ''m''=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
== References ==
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