1 | function C = adjoint(A, F) |
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2 | % ADJOINT Computes the adjoint of the quaternion matrix A. |
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3 | % |
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4 | % adjoint(A) or |
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5 | % adjoint(A, 'complex') returns a complex adjoint matrix. |
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6 | % adjoint(A, 'real') returns a real adjoint matrix. |
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7 | % |
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8 | % The definition of the adjoint matrix is not unique (several |
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9 | % permutations of the layout are possible). Note that if the |
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10 | % quaternion A is complexified, it is not possible to compute |
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11 | % a complex adjoint, since the complex values would have |
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12 | % complex real and imaginary parts. In this case, the real |
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13 | % adjoint will work, but its elements will be complex (!). |
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14 | |
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15 | % Copyright © 2005 Stephen J. Sangwine and Nicolas Le Bihan. |
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16 | % See the file : Copyright.m for further details. |
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17 | |
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18 | error(nargchk(1, 2, nargin)), error(nargoutchk(0, 1, nargout)) |
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19 | |
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20 | if nargin == 1 |
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21 | F = 'complex'; % Supply the default parameter value. |
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22 | end |
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23 | |
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24 | if ~strcmp(F, 'real') & ~strcmp(F, 'complex') |
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25 | error('Second parameter value must be ''real'' or ''complex''.') |
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26 | end |
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27 | |
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28 | % Extract the components of A. We use scalar() and not s() so that we |
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29 | % get zero if A is pure. |
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30 | |
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31 | R = scalar(A); X = x(A); Y = y(A); Z = z(A); |
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32 | |
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33 | if strcmp(F, 'complex') |
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34 | |
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35 | if all(all(imag(R) ~= 0)) | all(all(imag(X) ~= 0)) | ... |
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36 | all(all(imag(Y) ~= 0)) | all(all(imag(Z) ~= 0)) |
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37 | error('Cannot build a complex adjoint with complex elements: use ''real''.'); |
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38 | end |
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39 | |
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40 | % Reference: |
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41 | % |
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42 | % F. Z. Zhang, Quaternions and Matrices of Quaternions, |
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43 | % Linear Algebra and its Applications, 251, January 1997, 21-57. |
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44 | |
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45 | A1 = complex(R, X); A2 = complex(Y, Z); |
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46 | |
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47 | C = [[ A1, A2 ]; ... |
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48 | [-conj(A2), conj(A1)]]; |
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49 | |
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50 | else % F must be 'real', since we checked it above. |
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51 | |
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52 | % Reference: |
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53 | % |
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54 | % Todd A. Ell, 'Quaternion Notes', 1993-1999, unpublished, defines the |
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55 | % layout for a single quaternion. The extension to matrices of quaternions |
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56 | % follows easily in similar manner to Zhang above. |
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57 | % |
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58 | % An equivalent matrix representation for a single quaternion is noted |
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59 | % by Ward, J. P., 'Quaternions and Cayley numbers', Kluwer, 1997, p91. |
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60 | % It is the transpose of the representation used here. |
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61 | |
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62 | C = [[ R, X, Y, Z]; ... |
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63 | [-X, R, -Z, Y]; ... |
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64 | [-Y, Z, R, -X]; ... |
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65 | [-Z, -Y, X, R]]; |
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66 | end |
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67 | |
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