# Matrix expression question

• Sep 29th 2010, 09:00 PM
Matrix expression question
say $\displaystyle Y=[y_1,y_2,...y_k]$, where $\displaystyle y_j$ are column vectors. $\displaystyle a=[a_1,a_2...a_k]^t$ is a column vector. Now I have the a polynomial $\displaystyle \frac{1}{k}\sum_{j=1}^{k}a_j^2y_j^ty_j$. Is it possible to express this polynomial in a form of Y and a?
• Sep 30th 2010, 06:20 AM
HallsofIvy
Try $\displaystyle \frac{1}{k}a^t(y^ty)a$
• Sep 30th 2010, 06:54 AM
You mean $\displaystyle a^tY^tYa$? No, that's not right.
• Sep 30th 2010, 06:55 AM
Ackbeet

I'm not sure I agree that that will work.

$\displaystyle \displaystyle{\frac{1}{k}a^{t}(y^{t}y)a=\frac{1}{k }(y^{t}y)(a^{t}a)=\frac{1}{k}\left(\sum_{j}y_{j}^{ t}y_{j}\right)\left(\sum_{l}a_{l}^{t}a_{l}\right). }$

Is that necessarily equal to the desired expression? I'm not sure. I think you're going to have to use some sort of component-wise multiplication. Define the vector
$\displaystyle x\otimes y$ by

$\displaystyle (x\otimes y)_{j}=x_{j}y_{j}.$

Then you could write the desired expression as $\displaystyle \displaystyle{\frac{1}{k}(a\otimes a)\cdot(y\otimes y).}$
• Sep 30th 2010, 07:13 AM
I am not sure defining this vector can solve my problem. I wanted the polynomial in a matrix form because I need to do contained optimization with $\displaystyle a^ta=1$
• Sep 30th 2010, 07:14 AM
Ackbeet
Why don't you post your original question in full?
• Sep 30th 2010, 10:33 AM
Thank you for all your reply. I think I found a way to work around it. My problem is try to minimize the objective function$\displaystyle \frac{1}{k}\sum_{j=1}^k(Xw-y_ja_j)^t(Xw-y_ja_j)$. $\displaystyle w$ and $\displaystyle a=[a_j]$ are variable vectors. I tried to solve $\displaystyle w$ first then $\displaystyle a$, which gave me the question in this thread. Now I solve $\displaystyle a_j$ first, then $\displaystyle w$ has a closed form solution.
• Sep 30th 2010, 10:48 AM
Ackbeet
Well, if you've solved your problem, that's great! Have a good one.