Are there any canonical forms for tensors of type (2,1)? Such a tensor can be defined as a bi-linear map $$ T:V \times V \rightarrow V,$$ for $V$ a finitely dimensional real vector space.
2026-03-26 17:52:31.1774547551
Canonical forms for tensors of type (2,1)
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I think that in some sense there can't be any "worthwhile" canonical form:
If $V$ is $n$-dimensional then the space of such tensors is $n^3$-dimensional, and $GL(V)$ is only $n^2$-dimensional. So any canonical form would have $n^3-n^2$ degrees of freedom, which is almost as many as if you didn't use the canonical form.