Need to rewrite linear combination as vector expression.

[devonho]

Homework Statement

Given vectors
${\bf r}=\left[r_1,r_2,r_3\ldots{}r_n\right]^T$
${\bf e}=\left[e_1,e_2,e_3\ldots{}e_n\right]^T$

I need to write the sum

$\sum_{i=1}^{n}r_ie_i^2$

in terms of ${\bf r}$ and ${\bf e}$

Homework Equations

Nil.

The Attempt at a Solution

Without $r_i$, I am able to write $\sum_{i=1}^{n}e_i^2$ as

$\sum_{i=1}^{n}e_i^2={\bf e}^T{\bf e}$

If $r_i$ can be independent of $i$ I should be able to move it out of the summation. I am looking for some expansion/reexpression of $r_i$.

 

[tiny-tim]

hi devonho!

I need to write the sum

$\sum_{i=1}^{n}r_ie_i^2$

in terms of ${\bf r}$ and ${\bf e}$

why??

(i don't think you can)

 

[devonho]

This seems to work:

${\bf r}=<br /> \left[<br /> \begin{array}{ccccc}<br /> r_1 & 0 & \ldots & & 0\\<br /> 0 & r_2 & & & \vdots \\<br /> \vdots & & r_3 & & \\<br /> & & & \ddots & \\<br /> 0 & \ldots & & & r_n \\<br /> \end{array}<br /> \right]$

${\bf e}=<br /> \left[<br /> \begin{array}{c}<br /> e_1 \\<br /> e_2 \\<br /> \vdots\\<br /> e_n \\<br /> \end{array}<br /> \right]$

${\bf e}^T{\bf re}=\left[ e_1, e_2 \ldots e_n\right]<br /> \left[<br /> \begin{array}{ccccc}<br /> r_1 & 0 & \ldots & & 0\\<br /> 0 & r_2 & & & \vdots \\<br /> \vdots & & r_3 & & \\<br /> & & & \ddots & \\<br /> 0 & \ldots & & & r_n \\<br /> \end{array}<br /> \right]<br /> \left[<br /> \begin{array}{c}<br /> e_1 \\<br /> e_2 \\<br /> \vdots\\<br /> e_n \\<br /> \end{array}<br /> \right]<br /> =r_1e_1^2+r_2e_2^2\ldots +r_ne_n^2<br /> =\sum_{i=1}^{n}r_ie_i^2$

 

[tiny-tim]

but devonho, how do you form that middle matrix out of the vector r ?