Write $\sum a_nu_n ^2$ in matrix notation

Question

I just came across this seemingly simple question, but i am stumped.

Say I have this $ \sum u_n ^2 $ I know I can write this as u.u but what happens if I have $\sum a_nu_n ^2$ ? Is there a simple way to write this as a dot product?

To give some context, I was in a machine learning lecture and we were trying to calculate the weights for the linear regression, so minimizing $\sum (y_n - w^Tx_n)^2$. This went all fine and I got it correctly. Now the question I was posed was what happens when you want to add a cost to each weight so the function now becomes $ \sum c_n(y_n - w^Tx_n)^2$, and now suddenly I cannot write this in matrix notation.

Any help?

Answer 1

You have \begin{align} A = \begin{pmatrix} a_1 & 0 &\ldots & 0\\ 0 & a_2 & \ldots & 0\\ \vdots & \ldots & \ddots &\vdots\\ 0 & \ldots & \ldots & a_n \end{pmatrix} \ \ \text{ and } \ \ u = \begin{pmatrix} u_1 \\ u_2 \\ \vdots \\ u_n \end{pmatrix} \end{align} then it follows \begin{align} \sum^n_{i=1}a_nu_n^2=u^TAu. \end{align}

Answer 2

You could define the diagonal matrix $A:\{a_{ij}\}_{1\le i \le n, \; 1\le j \le n}=\delta_{ij}a_i$, i.e. the matrix with the vector $a$ on the main diagonal and zero in the other elements, then the product you are looking for is simply $u^TAu$.