Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

If $w_1$ and $w_2$ are strictly positive, show that the definition $$(x_1,x_2)\dot\ (y_1,y_2)=x_1y_1w_1+x_2y_2w_2,$$ yields an inner product on $\mathbb{R}^2$. Generalize this to $\mathbb{R}^p$.

I am confused on how to do this problem since it doesn't elaborate what $x_1,x_2,y_1,y_2$ are? For these problems should I automatically assume that they are vectors in the Cartesian space? Then I will prove it by letting $x_1=x_i$ and denoting $\vec{x_i}=x_1,...,x_n$ and then proving it by the definition, thus having $w_1(\vec{x_i}+\vec{y_i})=w_1(\langle x_1,...,x_n\rangle\dot\ \langle y_1,...,y_n \rangle)$ ... and preceding conventionally?

share|cite|improve this question
You need to find the list of properties of inner products from your text. Then show those properties are valid for any choice of $x_i$'s and $y_i$, and $w_i>0$. – Maesumi Feb 26 '13 at 4:05
The inner product that you have defined above is called a ‘weighted inner product’. – Haskell Curry Feb 26 '13 at 5:08
up vote 1 down vote accepted

You might find this description useful. The standard dot product in $\mathbb R^2$ can be written as a matrix product, with one of the two vectors transposed:

$$\vec{x}\cdot\vec{y} = \left(\begin{array}{cc} x_1 & x_2 \end{array}\right)\left(\begin{array}{c} y_1 \\ y_2 \end{array}\right)$$

The one you are dealing with is another valid inner product which is

$$w(\vec{x},\vec{y}) = \left(\begin{array}{cc} x_1 & x_2 \end{array}\right) \left(\begin{array}{cc} w_1 & 0 \\ 0 & w_2 \end{array}\right) \left(\begin{array}{c} y_1 \\ y_2 \end{array}\right).$$ In general, any expression that looks like this is an inner product when the matrix in the middle is positive definite. The generalization to $\mathbb R ^n$ is

$$\vec{x}\cdot\vec{y} = \left(\begin{array}{cccc} x_1 & x_2 & \cdots & x_n\end{array}\right) \left(\begin{array}{cccc} w_1 & 0 & 0 & 0 \\ 0 & w_2 & 0 & 0 \\ 0& 0& \ddots & \vdots \\ 0&0&\cdots & w_n \end{array}\right) \left(\begin{array}{c} y_1 \\ y_2 \\ \vdots \\y_n \end{array}\right)$$

where the $w_i$ have to satisfy the original conditions.

share|cite|improve this answer
Thank you for that. – Lays Feb 27 '13 at 0:23

You are absolutely perfect in assuming the $(x_1,x_2)\;\; and \;\; (y_1,y_2)$ are vectors in $\mathbb R^2$ You can then proceed conventionally - Proving all the properties of the inner product. I am slightly unclear what you intend to do for the second part of the problem.

IMO you have to then take a set $w_i\in \mathbb R : w_i>0 \;\forall\;i=1...p$ and get a generalization of the inner product defined to vectors in $\mathbb R^p$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.