I am trying to understand usage of skew matrices. So a few questions please:
1) It is my understanding that for vectors A and B, that A cross B is the same as the "skew symmetric matrix" of A times B. Is this correct?
2) If 1 is correct, I did a simple example in matlab where A = [1 0 0] and B = [0 1 0]. The obvious cross product is C = [0 0 1];
The skew multiplication is Sij = Ai*Bj - Aj*Bi which results in [0 1 0;-1 0 0; 0 0 0], but the skew of C, or AxB, is [0 -1 0; 1 0 0; 0 0 0]. Looks like the two results are negated or transposed. So I am not sure how this is equivalent, but maybe I just don't understand the application. Any explanation is appreciated.
3) If skew multiplication is a generalization to N dimensional vectors of the cross product when N = 3, then is there an example of finding an orthogonal vector to two vectors in 4-D? All the examples I can find show the equivalent skew matrix for a 3-D vector and how that multiplied with another 3-D vector results in the cross product vector, but I see no reference for getting the skew mapping for N dimensional vectors so that you can multiply that with another N-dimensional vector to obtain an orthogonal N-D vector.
Hope this is the correct forum.
Thanks in advance.
1) It is my understanding that for vectors A and B, that A cross B is the same as the "skew symmetric matrix" of A times B. Is this correct?
2) If 1 is correct, I did a simple example in matlab where A = [1 0 0] and B = [0 1 0]. The obvious cross product is C = [0 0 1];
The skew multiplication is Sij = Ai*Bj - Aj*Bi which results in [0 1 0;-1 0 0; 0 0 0], but the skew of C, or AxB, is [0 -1 0; 1 0 0; 0 0 0]. Looks like the two results are negated or transposed. So I am not sure how this is equivalent, but maybe I just don't understand the application. Any explanation is appreciated.
3) If skew multiplication is a generalization to N dimensional vectors of the cross product when N = 3, then is there an example of finding an orthogonal vector to two vectors in 4-D? All the examples I can find show the equivalent skew matrix for a 3-D vector and how that multiplied with another 3-D vector results in the cross product vector, but I see no reference for getting the skew mapping for N dimensional vectors so that you can multiply that with another N-dimensional vector to obtain an orthogonal N-D vector.
Hope this is the correct forum.
Thanks in advance.
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