how to compute explicitly the cross product of a matrix with a vector

Question

I have been working on a physics problem, and at certain point, a book (Classical Electrodynamics, Jackson, 3rd ed, problem 6.10) defines the cross product of a matrix with a vector as follows:

$$\overleftrightarrow{M} = \overleftrightarrow{T} \times \vec{x}$$

In dyadic notation. If I am right, for three dimensions $\overleftrightarrow{T}$ and $\vec{x}$ can be written explicitly as follows:

$$ \overleftrightarrow{T} = \left( \begin{array}{lcr} T_{11} & T_{12} & T_{13} \\ T_{21} & T_{22} & T_{23} \\ T_{31} & T_{32} & T_{33} \\ \end{array} \right) $$ $$ \vec{x} = \left( \begin{array}{lcr} x_{1} \\ x_{2} \\ x_{3} \\ \end{array} \right) $$

How can I compute in an explicit manner $\overleftrightarrow{M}$? The cross product of $\overleftrightarrow{T}$ and $\vec{x}$. I can't figure it out.

Does this product obeys the same rules that the cross product of 2 vectors? particularly,

$$-\vec{x} \times \overleftrightarrow{T} = \overleftrightarrow{T} \times \vec{x}$$

is it correct?

If you need more information, please let me know.

Answer 1

Given any $n \times n$ matrix $A$, one can decompose it as a sum of outer product of n pairs of vectors ($n \times 1$ matrices) $u_i$, $v_i$. $$A = \sum_{i=1}^n u_i \otimes v_i$$

For example, if $e_i$ is the $n \times 1$ matrix with a $1$ at row $i$ and $0$ elsewhere, following two choices of $u_i, v_i$ will reproduce $A$:

1. $u_i = e_i$ and $v_i = i^{th}$ row vector of $A$.
2. $u_i = i^{th}$ column vector of $A$ and $v_i = e_i$.

When $n = 3$, the cross product between $A$ and vector $w$ can be defined through this decomposition: $$\begin{align} A \times w &\stackrel{def}{=} \sum_{i=1}^3 u_i \otimes (v_i \times w)\\ w \times A &\stackrel{def}{=} \sum_{i=1}^3 (w \times u_i ) \otimes v_i \end{align} $$

In terms of components, you have $$\begin{align} ( A \times w )_{ab} &= \sum_{i=1}^n u_{i,a} (v_i \times w)_b = \sum_{i=1}^n u_{i,a} \left(\sum_{c=1}^3\sum_{d=1}^3 \varepsilon_{bcd} v_{i,c} w_d\right) = \sum_{c=1}^3\sum_{d=1}^3 \varepsilon_{bcd} A_{ac} w_d\\ ( w \times A )_{ab} &= \sum_{i=1}^n (w \times u_i )_a v_{i,b} = \sum_{i=1}^n \left(\sum_{c=1}^3\sum_{d=1}^3 \varepsilon_{acd} w_c u_{i,d}\right) v_{i,b} = \sum_{c=1}^3\sum_{d=1}^3 \varepsilon_{acd} w_c A_{db} \end{align} $$ where $\varepsilon_{bcd}$ is the Levi-Civita symbol for $3$-dim.

In general, $w \times A \ne - A \times w$. Instead, we have

$$w \times A^T = - (A \times w)^T$$

To compute $A \times w$, replace all row vectors $v_i$ by corresponding $v_i \times w$. Similarly, to compute $w \times A$, replace all column vectors $u_i$ by corresponding $w \times u_i$.