Ordinary matrix-vector multiplication

[Treadstone 71]

Let P=(p_{ij}) be a real symmetric 2x2 matrix. Show that the function on \mathbb{R}^2\times\mathbb{R}^2 (Where R^2 is a space of column vectors) defined by <v,w>=v^tPw is an inner product if and only if p_{11} and det(P) are both swtrictly positive.

I just need to know what Pw means in <v,w>=v^tPw.

 

[arildno]

Ordinary matrix-vector multiplication.

 

[Treadstone 71]

I see. I thought it was "P of w".

 

[Treadstone 71]

This function doesn't send vectors to scalars, it can't be an inner product, unless I understood something wrong.

 

[Treadstone 71]

v and w are 1x2 column vectors, right? So why is this function on R^2xR^2? Isn't it defined on the vector space R^2?

 

[shmoe]

v and w are 1x2 column vectors, right? So why is this function on R^2xR^2? Isn't it defined on the vector space R^2?

they're 2x1, but there are two of them. This function takes a pair of vectors, (v,w) and gives a real number, so the domain is R^2xR^2.

 

[HallsofIvy]

v is in R², w is in R² so (v, w) is in R² x R².