Matrix expression question

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

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

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

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
$x\otimes y$ by

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

Then you could write the desired expression as $\displaystyle{\frac{1}{k}(a\otimes a)\cdot(y\otimes y).}$
• September 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 $a^ta=1$
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 $\frac{1}{k}\sum_{j=1}^k(Xw-y_ja_j)^t(Xw-y_ja_j)$. $w$ and $a=[a_j]$ are variable vectors. I tried to solve $w$ first then $a$, which gave me the question in this thread. Now I solve $a_j$ first, then $w$ has a closed form solution.