I just finished reading through Introduction to Matrices and Vectors, International Student Edition, by Jacob T. Schwartz. In chapter 6 they proposed a definition I've never seen before, that seems uncannily similar to the cross product of two vectors.

Jacob starts with explaining that there is a one-to-one correspondance between skew matrices and 3-vectors.

$$\begin{bmatrix} 0 & u & v\\ -u & 0 & w \\ -v & -w & 0 \end{bmatrix} \leftrightarrow \begin{bmatrix} u\\ v\\ w \end{bmatrix}$$

He employs the following definition of a cross-product of a difference in products between two matrices multiplied in the standard way:

$$A \times B = AB - BA$$

He proves a theorem that if $A$ and $B$ are skew matrices, then $A \times B$ is also a skew matrix. Combining the one-to-one correspondance with this theorem, he defines a cap product between two any 3-vectors by converting them both to skew matrices, taking the cross-product of the skew matrices, and converting the resulting skew matrix back into a vector. He goes on to prove a componentwise calculation of this product to be the following:

$$U, V \in \mathbb{R}^3$$ $$U \land V = \begin{bmatrix} u_2v_3 - v_2 u_3\\ u_3 v_1 - v_3 u_1\\ u_1 v_2 - v_1 u_2 \end{bmatrix}$$

Looking at the component-wise formula, this appears to be the same the standard cross product of vectors, however Jacob describes an orientation rule where $U \land V$ would have the hand straight with the fingers representing $U$, the thumb pointing up representing $V$, and the direction of the palm representing $U \times V$. This sounds like the negative direction of the cross-product...

Is this cap product of vectors just the cross-product of vectors in disguise? or is it something else?


1 Answer 1


When I was an undergraduate, more than half a century ago, I struggled with the definition of cross-product in terms of bending my hand into some unnatural position and trying to deduce something from the direction in which my thumb and index finger were pointing. It is sad to see this practtice still going on, with students making weird gestures in examinations, thereby attracting unwelcome attention from invigilators. My advice is to take the algebraic formula you gave as the $definition$ of cross-product. That definition makes the 3-component vectors into a Lie algebra with the Lie bracket as the cross-product. Another Lie algebra is the set of all $n \times n$ matrices with the Lie bracket as the commutator. Then the skew-symmetric matrices form a sub-algebra and what Jacob is doing is pointing out that that sub Lie algebra , when $n=3,$ is isomorhic to the Lie algebra of 3-vectors with the Lie bracket as the cross-product.


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