Question on Clifford Algebra

dimension10

I was trying to solve the following equation:

[tex] \bigwedge\limits_{j=1}^{k}\begin{bmatrix}
a_{1,j}\\
a_{2,j}\\
:\\
.\\
a_{k+1,j}
\end{bmatrix} [/tex]

Does anyone know how I can solve it? Thanks in advance.

algebrat

try k=1,2,3 etc.

dimension10

Actually, I already know what that expression results in. I was trying to prove it.

algebrat

There is no equation, just an expression. It is not clear what you are trying to solve.

Please show us the result. Please tell us exactly what you are trying to prove.

dimension10

Ya I realised that later. Just made a typo error.



[tex]\sqrt {\sum\limits_{j = 1}^{k + 1} {{{\det }^2}\left( {{{{\mathop{\rm cross}\nolimits} }_j}\left( {\mathop {\bf{A}}\limits_{(k + 1) \times k} } \right)} \right)} } [/tex]

where A_(k+1)Xk is the matrix formed by augmenting the vectors together and the cross_j function means crossing out the j^th row of the matrix A_(k+1)Xk.

I'm trying to prove that [itex]\sqrt {\sum\limits_{j = 1}^{k + 1} {{{\det }^2}\left( {{{{\mathop{\rm cross}\nolimits} }_j}\left( {\mathop {\bf{A}}\limits_{(k + 1) \times k} } \right)} \right)} } [/itex] is the answer, since I found it by finding the special cases where k=1,2,3.

algebrat

I guess I know very little about clifford algebras, I was expecting a wedge not to be a scalar, which I am guessing is what the root of sum of determinants squared would give.

dimension10

Oops! Sorry, you are right. I mean the determinant of the exterior product is equal to [tex]\sqrt {\sum\limits_{j = 1}^{k + 1} {{{\det }^2}\left( {{{{\mathop{\rm cross}\nolimits} }_j}\left( {\mathop {\bf{A}}\limits_{(k + 1) \times k} } \right)} \right)} }[/tex]

algebrat

For k=2 I would have guessed the cross product.

Ah

dimension10

No... I think it is the hodge dual of the cross product, actually. The wedge alone would be a bivector in that case.

algebrat

Show us the steps for k=2 and/or 3

dimension10

[tex]\begin{array}{l}
\begin{array}{*{20}{l}}
{\left\| {\left[ {\begin{array}{*{20}{l}}
\alpha \\
\gamma \\
\varepsilon
\end{array}} \right] \wedge \left[ {\begin{array}{*{20}{l}}
\beta \\
\delta \\
\zeta
\end{array}} \right]} \right\| = \left\| {\alpha \delta \left( {{{{\bf{\hat e}}}_1} \wedge {{{\bf{\hat e}}}_2}} \right) + \alpha \zeta \left( {{{{\bf{\hat e}}}_1} \wedge {{{\bf{\hat e}}}_3}} \right) + \gamma \beta \left( {{{{\bf{\hat e}}}_2} \wedge {{{\bf{\hat e}}}_1}} \right) + \gamma \zeta \left( {{{{\bf{\hat e}}}_2} \wedge {{{\bf{\hat e}}}_3}} \right) + \varepsilon \beta \left( {{{{\bf{\hat e}}}_3} \wedge {{{\bf{\hat e}}}_1}} \right) + \varepsilon \delta \left( {{{{\bf{\hat e}}}_3} \wedge {{{\bf{\hat e}}}_2}} \right)} \right\|}\\
{ = \left\| {\left( {\alpha \delta - \beta \gamma } \right)\left( {{{{\bf{\hat e}}}_1} \wedge {{{\bf{\hat e}}}_2}} \right) + \left( {\alpha \zeta - \beta \varepsilon } \right)\left( {{{{\bf{\hat e}}}_1} \wedge {{{\bf{\hat e}}}_3}} \right) + \left( {\gamma \zeta - \delta \varepsilon } \right)\left( {{{{\bf{\hat e}}}_2} \wedge {{{\bf{\hat e}}}_3}} \right)} \right\|}\\
{ = \left\| {\det \left[ {\begin{array}{*{20}{c}}
\alpha &\beta \\
\gamma &\delta
\end{array}} \right]\left( {{{{\bf{\hat e}}}_1} \wedge {{{\bf{\hat e}}}_2}} \right) + \det \left[ {\begin{array}{*{20}{c}}
\alpha &\beta \\
\varepsilon &\zeta
\end{array}} \right]\left( {{{{\bf{\hat e}}}_1} \wedge {{{\bf{\hat e}}}_3}} \right) + \det \left[ {\begin{array}{*{20}{c}}
\gamma &\delta \\
\varepsilon &\zeta
\end{array}} \right]\left( {{{{\bf{\hat e}}}_2} \wedge {{{\bf{\hat e}}}_3}} \right)} \right\|}
\end{array}\\
= \sqrt {{{\det }^2}\left[ {\begin{array}{*{20}{c}}
\alpha &\beta \\
\gamma &\delta
\end{array}} \right] + {{\det }^2}\left[ {\begin{array}{*{20}{c}}
\alpha &\beta \\
\varepsilon &\zeta
\end{array}} \right] + {{\det }^2}\left[ {\begin{array}{*{20}{c}}
\gamma &\delta \\
\varepsilon &\zeta
\end{array}} \right]}
\end{array}[/tex]

[tex]\begin{array}{l}
\left\| {\left[ \begin{array}{l}
\alpha \\
\delta \\
\eta \\
\kappa
\end{array} \right] \wedge \left[ \begin{array}{l}
\beta \\
\varepsilon \\
\theta \\
\lambda
\end{array} \right] \wedge \left[ \begin{array}{l}
\gamma \\
\zeta \\
\iota \\
\mu
\end{array} \right]} \right\| = \left\| {\left( {\alpha {{{\bf{\hat e}}}_1} + \delta {{{\bf{\hat e}}}_2} + \eta {{{\bf{\hat e}}}_3} + \kappa {{{\bf{\hat e}}}_4}} \right) \wedge \left( {\beta {{{\bf{\hat e}}}_1} + \varepsilon {{{\bf{\hat e}}}_2} + \theta {{{\bf{\hat e}}}_3} + \lambda {{{\bf{\hat e}}}_4}} \right) \wedge \left( {\gamma {{{\bf{\hat e}}}_1} + \zeta {{{\bf{\hat e}}}_2} + \iota {{{\bf{\hat e}}}_3} + \mu {{{\bf{\hat e}}}_4}} \right)} \right\|\\
{\rm{ }} = \left| {\left| {\left( {\alpha \varepsilon \iota - \delta \beta \iota - \alpha \theta \zeta + \eta \beta \zeta + \delta \theta \gamma - \eta \varepsilon \gamma } \right)\left( {{{{\bf{\hat e}}}_1} \wedge {{{\bf{\hat e}}}_2} \wedge {{{\bf{\hat e}}}_3}} \right) + } \right.} \right.\\
{\rm{ }}\left( {\alpha \varepsilon \mu - \delta \beta \mu - \alpha \lambda \zeta + \kappa \beta \zeta - \delta \lambda \gamma + \kappa \varepsilon \gamma } \right)\left( {{{{\bf{\hat e}}}_1} \wedge {{{\bf{\hat e}}}_2} \wedge {{{\bf{\hat e}}}_4}} \right) + \\
{\rm{ }}\left( {\alpha \theta \mu - \eta \beta \mu - \alpha \lambda \iota + \kappa \beta \iota - \eta \lambda \gamma + \kappa \theta \gamma } \right)\left( {{{{\bf{\hat e}}}_1} \wedge {{{\bf{\hat e}}}_3} \wedge {{{\bf{\hat e}}}_4}} \right) + \\
\left. {\left. {{\rm{ }}\left( {\delta \theta \mu - \eta \varepsilon \mu - \delta \lambda \iota + \kappa \varepsilon \iota - \eta \lambda \zeta + \kappa \theta \zeta } \right)\left( {{{{\bf{\hat e}}}_2} \wedge {{{\bf{\hat e}}}_3} \wedge {{{\bf{\hat e}}}_4}} \right)} \right|} \right|\\
{\rm{ }} = \left| {\left| {\det \left[ {\begin{array}{*{20}{c}}
\alpha &\beta &\gamma \\
\delta &\varepsilon &\zeta \\
\eta &\theta &\iota
\end{array}} \right]\left( {{{{\bf{\hat e}}}_1} \wedge {{{\bf{\hat e}}}_2} \wedge {{{\bf{\hat e}}}_3}} \right) + } \right.} \right.\\
{\rm{ }}\det \left[ {\begin{array}{*{20}{c}}
\alpha &\beta &\gamma \\
\delta &\varepsilon &\zeta \\
\kappa &\lambda &\mu
\end{array}} \right]\left( {{{{\bf{\hat e}}}_1} \wedge {{{\bf{\hat e}}}_2} \wedge {{{\bf{\hat e}}}_4}} \right) + \\
{\rm{ }}\det \left[ {\begin{array}{*{20}{c}}
\alpha &\beta &\gamma \\
\eta &\theta &\iota \\
\kappa &\lambda &\mu
\end{array}} \right]\left( {{{{\bf{\hat e}}}_1} \wedge {{{\bf{\hat e}}}_3} \wedge {{{\bf{\hat e}}}_4}} \right) + \\
\left. {\left. {{\rm{ }}\det \left[ {\begin{array}{*{20}{c}}
\delta &\varepsilon &\zeta \\
\eta &\theta &\iota \\
\kappa &\lambda &\mu
\end{array}} \right]\left( {{{{\bf{\hat e}}}_2} \wedge {{{\bf{\hat e}}}_3} \wedge {{{\bf{\hat e}}}_4}} \right)} \right|} \right|\\
{\rm{ }} = \sqrt {{{\det }^2}\left[ {\begin{array}{*{20}{c}}
\alpha &\beta &\gamma \\
\delta &\varepsilon &\zeta \\
\eta &\theta &\iota
\end{array}} \right] + {{\det }^2}\left[ {\begin{array}{*{20}{c}}
\alpha &\beta &\gamma \\
\delta &\varepsilon &\zeta \\
\kappa &\lambda &\mu
\end{array}} \right] + {{\det }^2}\left[ {\begin{array}{*{20}{c}}
\alpha &\beta &\gamma \\
\eta &\theta &\iota \\
\kappa &\lambda &\mu
\end{array}} \right] + {{\det }^2}\left[ {\begin{array}{*{20}{c}}
\delta &\varepsilon &\zeta \\
\eta &\theta &\iota \\
\kappa &\lambda &\mu
\end{array}} \right]}
\end{array}[/tex]

But when I tried it for the general case, it was not possible.

algebrat

maybe you could somehow argue that the coefficient of e_1^...^e_{j-1}^e_{j+1}^...^e_{k+1} would be the det of cross_j(A).

dimension10

Thanks a lot! I think that you're right! Thanks again!