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.