Lagmental Vicfred

Exterior Powers Record All Minors at Once by Vicfred

Last updated: Fri 17 March 2017

tags: mathematics, multilinear algebra, tensor products, matrix functions

The kth exterior power of a linear map has matrix entries equal to its k by k minors. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

Notation

Tensor and exterior powers turn multilinear behavior into linear maps. For finite-dimensional \(V\), the spaces \(V^{\otimes k}\) and \(\bigwedge^kV\) carry induced actions of every \(T\in\operatorname{End}(V)\).

$$ \bigwedge\nolimits^kT:\bigwedge\nolimits^kV\to\bigwedge\nolimits^kV $$

I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.

$$ (\bigwedge\nolimits^kT)_{I,J}=\det(T_{I,J}) $$

Stress the formula

Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.

$$ Te_{j_1}\wedge\cdots\wedge Te_{j_k}=\sum_{i_1<\cdots<i_k}\det(T_{(i_1,\ldots,i_k),(j_1,\ldots,j_k)})\,e_{i_1}\wedge\cdots\wedge e_{i_k} $$

Interpretation

The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.

$$ \begin{aligned} \mathsf{D}\;&:\quad \bigwedge\nolimits^kT:\bigwedge\nolimits^kV\to\bigwedge\nolimits^kV,\\[5pt] \mathsf{C}\;&:\quad (\bigwedge\nolimits^kT)_{I,J}=\det(T_{I,J}). \end{aligned} $$

Limit of the argument

Tensor coordinates depend on a basis even when the tensor does not. Index notation is safe only when contraction rules and variance are clear.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] (\bigwedge\nolimits^kT)_{I,J}=\det(T_{I,J}) \end{gathered}} $$

With the dependency made explicit, the same pattern can be recognised safely in nearby problems. A changed hypothesis should now be easy to spot.

This article was posted on Sat 26 March 2011. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.