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Tensors
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this post was submitted on 03 Sep 2024
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Square matrices are linear endomorphisms. They are isomorphic to (1,1) tensors but not any other rank of tensors.
a tensor is a multi-linear map V × ... × V × V^^ × ... × V^^ → F, and a multi-linear map V × ... × V × V^^ × ... × V^^ → F is the same as a linear map V ⊗ ... ⊗ V ⊗ V^^ ⊗ ... ⊗ V^^ → F. and a linear map is ""the same thing as"" a matrix. so in this way, you can associate matrices to tensors. (but the matrices are formed in the tensor space V ⊗ ... ⊗ V ⊗ V^^ ⊗ ... ⊗ V^^, not in the vector space V.)