3
$\begingroup$

(deprecated-taken back based on discussion(OLD)) What is a good way to factor a symmetric matrix $X$ as an outer product of two vectors $u$ and $v$. i.e, Find two vectors $u$ and $v$ such that $X=uv^T$, where $X$ is a symmetric matrix.

(Updated/ New Question of interest(NEW)) Given a symmetric matrix $X$, what is a way to figure out the best possible vectors $u$ and $v$ such that the error under say an l2 loss over $X-uv^T$ is minimum. Feel free to make notes about any optimality conditions/ assumptions that might go around this problem.

  • 0
    @hardmath Question re-edited.Thanks for the direction.2012-01-25

1 Answers 1

6

Since $X$ is symmetric it is always possible to put it in diagonal form using an orthonormal basis, i.e., $X = UDU^T$, where $D = \operatorname{diag}(\lambda_1, \lambda_2, \dots, \lambda_n)$ are the eigenvalues of $X$. From this it is possible to see that $X = \sum_{i = 1}^n \lambda_i u_i u_i^T$ where $u_i$ is the $i$:th column of $U$.

Now, assuming the eigenvalues are ordered by absolute value, $|\lambda_1| \geq |\lambda_2| \geq \dots \geq |\lambda_n|$, the best approximation $\tilde X = u v^T$, in the sense that $\|X - \tilde X\|_2$ is minimized ($\| \cdot \|_2$ is the Frobenius norm) is given by $\tilde X = \lambda_1 u_1 u_1^T$ (you can of course write $v = \lambda_1 u_1$ if you want).

  • 0
    @ all. Thank you. Do look at the edited question in http://math.stackexchange.com/questions/102321/suitable-loss-function-for-order-preserving-factoring-of-a-matrix . It is in a similar setting, but the question deals with a suitable loss function- for the problem. Thank you.2012-01-26