Function notation and inner product

[JerryG]

Homework Statement

I have this problem, but I'm not familiar with the function notation that the professor is using. Can anyone tell me what is actually being asked? I understand everything up to the part that is in bold, but after that, I am lost.

Let A be a nxn matrix of real numbers, such that A can be written as a product of
another matrix B and its transpose (e.g. A=BT*B). Assuming that B is
nonsingular, show that the function d:RnxRn->R, d(x,y)=xTAy is a vector dot
product.

Some of the formating was lost so Rn is shown as R^n and xT is x transpose.

 

[LCKurtz]

x and y are in Rⁿ, so they are n by 1 matrices so x^TAy is (1 by n)(n by n)(n by 1) = 1 by 1 or scalar. If you put in A = B^TB I think you will see that it is a dot product of two vectors if you look at it right.

 

[Fredrik]

I would interpret the problem as "Show that d satisfies the definition of an inner product". This is really easy if you know the definition and you're comfortable with matrix algebra (stuff like $(XY)^T=Y^T X^T$).

If you're only supposed to show that $x^TAy$ is a dot product of two vectors, the complete solution would be $x^TAy=x^T(Ay)$, because Ay is a column vector. (You know that the definition of matrix multiplication implies that $u^Tv=u_1v_1+\dots+u_nv_n$ when u and v are column matrices, right?)