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I'm trying to figure how, if ever possible, to optmise mathematically a problem, which I will describe hereafter.

I read similar questions about this topic, but I ask the question nevertheless since this case might have some special properties for which I can do something about.

I might be wrong, but I have the feeling there would be a way, but I don't have the competencies to formalise it in a mathematical way. So, here I am.

Let $A(\omega) = C(\omega) \odot S(\omega)$, where $C(\omega) = \hat{C}^{\omega}$, $\hat{C} \in \mathbb{R}^{n \times n}$ being a symmetric, positive definite matrix, $\omega \in \mathbb{R}$, and $S(\omega)$ being the outer product of the vector $s(\omega) \in \mathbb{R}^{n \times 1}$, i.e. $S(\omega) = s(\omega) \otimes s(\omega)$. We can also say that $A \in \mathbb{R}^{n \times n}$ is symmetric and positive-definite.

NOTE: $\odot$ denotes Hadamard product (i.e. element-wise product), while $C(\omega) = \hat{C}^{\omega}$ denotes matrix power (to the power $\omega$, i.e. $c_{ij}(\omega) = {\hat{c}_{ij}}^{\omega}$).

I would like to find an easier way to compute $SVD(A(\omega))$, possibly avoid it. Why? Because I could compute $\hat{C} = U_1 \Sigma_1 U_1^T$ once and forall, and then probably having that $C(\omega) = U_1 {\Sigma_1}^{\omega} U_1^T$, as well as $S(\omega) = \mu(\omega) \ \sigma \ \mu(\omega)^T$ where $\mu(\omega) = s(\omega) / ||s(\omega)||$ and $\sigma = ||s(\omega)||^2$. So, I would like to find a way to compute SVD$(A(\omega) = U_1 {\Sigma_1}^{\omega} U_1^T \odot \mu(\omega) \mu(\omega)^T)$. I wonder if this cannot be related to something like $A(\omega) = U \Sigma U^T$, where $U = U_1 \odot \mu(\omega)$ and $\Sigma = \sigma \Sigma_1$, which would then just require two multiplications to have the equivalent SVD of matrix $A(\omega)$.

I'm open to listen to all suggestions/criticisms/lessons on the topic. Thanks to anyone answering, and sorry if I might sound stupid.

mEm
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  • Could you clarify what you mean by $\odot$ and $\hat{C}^{\omega}?$ – Ragon Jun 29 '23 at 15:39
  • @Ragon I edited the question, thanks – mEm Jun 29 '23 at 16:14
  • Let $D=\operatorname{Diag}(s),,$ then the Hadamard product can be replaced as follows $$\eqalign{ A &= \left(U\Sigma^w V^T\right)\odot ss^T \ &= D\left(U\Sigma^w V^T\right)D \ }$$ – greg Jul 01 '23 at 00:19
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    Thanks @greg , very useful. However, I fould out that another problem lies in the relationship $$C(\omega) = \hat{C}^{\omega} = U_1 {\Sigma_1}^{\omega} U_1^T$$ which is true i.f.f. $$\omega \in \mathbb{N}$$ is a natural number, not a real one.. So I cannot apply that relationship straight ahead, which complicates things quite a lot. – mEm Jul 09 '23 at 13:56

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