On Monday, November 17, 2025 at 9:54:00 PM UTC-7 Russell Standish wrote:
On Sat, Nov 15, 2025 at 08:22:23PM -0800, Alan Grayson wrote: > > > On Friday, November 14, 2025 at 8:30:29 PM UTC-7 Alan Grayson wrote: > > On Friday, November 14, 2025 at 7:42:23 PM UTC-7 Russell Standish wrote: > > T(u,v) = uᵀMv where M is the matrix representation of T, and ᵀ is the > transpose operator. > > > This is too succinct for me to understand your explanation. AG > > > Is the above how a tensor is generally evaluated? How would it be evaluated if > T has three independent variables, u,v,w? AG > > If u and v are modeled as row matrices, then u transposed is a colum vector, > and > the total result is a real number. But I don't think it's easy to show that > this is the > same value obtained by modeling the tensor as a linear function of u and v. > Offhand. > do you have a link for showing the equivalence? Further, in the case of T(u), > how is > the result a real number when using matrix notation? It looks like a vector > when u > is modeled as a column vector. OTOH, I don't think we can model u as a row > vector > to do the calculation in this situation (or can we?). AG > This is Linear Algebra 101. To convince yourself, try it with a 2x2 matrix to make the calculations easier. Extending the result to Rⁿis not difficult. Exercise: Show that (auᵀ₁+buᵀ₂)Mv = auᵀ₁Mv + buᵀ₂Mv and uᵀM(av₁+bv₂)=auᵀMv₁+ buᵀMv₂ for a,b∈R, uᵢ,vᵢ ∈ Rⁿ, and M a real valued matrix. The above two lines are the definition of a bilinear function from Rⁿ⟶ R. *If they're definitions, there's nothing to be shown or proven. But I have two* *questions relating to this subject. First, since uᵀ is a column matrix, is it OK* *to place it on the RHS of M, with the convention that Muᵀ is evaluated first, * *followed by the result being evaluated by applying v (a row matrix), so we* *we get a real constant as the result? Second, if the function has one * *independent variable, say u, I don't see how we can use matrix notation to* *evaluate the tensor To get a real value as the result. TY, AG* For a trilinear function, it is more convenient to use the Einstein summation convention, but basically it works the same way. ---------------------------------------------------------------------------- Dr Russell Standish Phone 0425 253119 (mobile) Principal, High Performance Coders [email protected] http://www.hpcoders.com.au ---------------------------------------------------------------------------- -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion visit https://groups.google.com/d/msgid/everything-list/03af535a-9f64-4bc7-9a9a-8a307e68cef0n%40googlegroups.com.
