What is a Tensor Product and How Does it Relate to Vectors and Matrices?

ngc_1729
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Hi, can anyone please explain me how to understand this term? I tried to expand it, but seems I may not be right, so can anyone help me with expasion of this rhs term below? T is suppsoed to be symmetric, but when I expand it it doesn't seem to be symmetric, please help.

consider 2 mutually orthogonal directions a1,a2. associated with sides of a rectangular plane whose sides are d1,d2. and this rectangular plane is oriented at an arbitrary angle wrt global x axis.
Now consier a Transformation T as a function of (a1,a2) and (d1,d2) as :
T = \Sigma[(1/di)ai X ai] where i =1 to 2 and X is tensor product

when I expanded rhs of the above experssion I got:

T11 = a1 d1/d1 , T12 = a1d2/d1, T21 = a2d1/d2 , T22 = a2d2/d2
am I correct? if I am why is this not symmetric?
 
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