4 ms·

Two interpretations of transpose: - Let <-,-> denote an inner product. Then <Au, v> = <u, A^T v> is a definition of the transpose operation. Proof: (Au)^T v

Two interpretations of transpose: - Let <-,-> denote an inner product. Then <Au, v> = <u, A^T v> is a definition of the transpose operation. Proof: (Au)^T v = u^T A^T v = <u, A^T v> - The transpose maps a vector v to a linear functional (a kind of function) v^T, such that v^T (u) = <v, u>. Conversely, a linear functional of a vector can be transposes to a vector. In fact, it's the latter (linear functionals to vectors) that's more interesting. This fact generalises to Hilbert spaces as the Riesz representation theorem, where the consequences can be pretty amazing. - An arbitrary linear operator L: H -> K tranposes to a mapping between their dual spaces L^* = K^* -> H^* by L*(f) = f o L. - The outer product produces rank-one matrices, and all rank-one matrices can be expressed as outer products.

geokon 8y ago
Oh yikes. So the rabbit holes goes quite deep So the transpose is really ties to the whole idea of inner product operators/function? (I'm not sure what the exact terminology is there) So thinking out loud a bit about the first interpretation... The inner to me represents an operator that maps vectors to a scalar that denotes the degree of "sameness" between two vectors. And it's a whole family of operators that all share certain properties. It just so happens that to write out this operator in matrix notation (for the standard inner product) we need to transpose/rotate the first vector which is yucky and why we prefer the bracket notation It's a cool insight but it just makes it seem like a mechanical operator that flips a column into a row to make things work for one family of operators. I'm thinking more like in the case of an orthonormal matrix/basis. We know the transpose is it's inverse. It stems from the standard inner products of the columns being equal to one which is I find kinda interesting b/c the inner products suddenly become part of linear systems and not just a separate measurement of a degree of sameness.. but the headache for me is that I can picture an orthonormal basis, but I can't picture it's inverse! :) PS: The orthonormal basis case and the transpose by extension further makes me feel that only the standard inner product seems to really map to something directly applicable in linear systems PPS: The 3rd bullet is a bit beyond me. And the 4th is also very interesting but I'm not sure how to expand on that to build an intuition haha. I'll need to digest it further

geokon 8y ago

Oh yikes. So the rabbit holes goes quite deep So the transpose is really ties to the whole idea of inner product operators/function? (I'm not sure what the exact terminology is there) So thinking out loud a bit about the first interpretation... The inner to me represents an operator that maps vectors to a scalar that denotes the degree of "sameness" between two vectors. And it's a whole family of operators that all share certain properties. It just so happens that to write out this operator in matrix notation (for the standard inner product) we need to transpose/rotate the first vector which is yucky and why we prefer the bracket notation It's a cool insight but it just makes it seem like a mechanical operator that flips a column into a row to make things work for one family of operators. I'm thinking more like in the case of an orthonormal matrix/basis. We know the transpose is it's inverse. It stems from the standard inner products of the columns being equal to one which is I find kinda interesting b/c the inner products suddenly become part of linear systems and not just a separate measurement of a degree of sameness.. but the headache for me is that I can picture an orthonormal basis, but I can't picture it's inverse! :) PS: The orthonormal basis case and the transpose by extension further makes me feel that only the standard inner product seems to really map to something directly applicable in linear systems PPS: The 3rd bullet is a bit beyond me. And the 4th is also very interesting but I'm not sure how to expand on that to build an intuition haha. I'll need to digest it further