I have some trouble with notation of bilinear functions. I will state the theore

Question

I have some trouble with notation of bilinear functions. I will state the theorem i have trouble with:

A function Rm x Rn   ---> R is bilinear if and only if it can be written in the form y=x1'Ax2 with A in Mmxn

Now, i will explain what i think is going on here. So I think what is meant with Rm x Rn ---> R is that a function whose domain is two arbitrarily vectors of dimension m and n respectively is mapped to a real number. For this to hold it should be possible to write it in y=x1'Ax2 where y is a real number x1' is a transposed vector and x2 is also a vector.

Now is this interpretation correct, that is, does Rm x Rn represent any two arbitrary vectors as input, and is the output of this function a real number? Also, Rm and Rn specify nothing with respect to the vector being a column or row vector right?

If any of my interpretation is wrong please tell me, and possibly make things clearer for me.

Thanks in advance

Explanation / Answer

Your interpretation is correct. The function takes two vectors as input, one of them(x1) is in R_m and the other(i.e x2) is in R_n and gives ouput a real number. x1 and x2 are column vectors. So x1 has size mx1 and x2 has size nx1. Also x1' has size 1xm. A has size mxn. So as a result, x1'Ax2 will give a real number

Related Questions

Navigate

• Browse (All)
• Browse I
• Subjects