say I have a $n\times n$ matrix $w_{ij}$. I can preform a singular value decomposition such that $w_{ij}=\sum_l \sum_n u_{il}\lambda_{ln}v_{jn}$ with $\lambda_{ln}$ diagonal. Now, is there such a generalization so that, given a function of two variables $w(\theta_1,\theta_2)$ such that $$ w(\theta_1 ,\theta_2)=\int dy \int dx \,u(\theta_1 ,x) \, \lambda(x,y) \, v(\theta_2 ,y) $$ where $\lambda$ plays a similar role like it did in the SVD? For instance, say I have the following $$ \exp {[\alpha \cos(\theta-\phi)]} $$ is it possible to find a decomposition such that $$ \exp {\alpha \cos(\theta-\phi)}=\int\int dx \, dy \, u(\theta,x) \, \lambda(\alpha,x,y) \, v(\phi,y) $$ Thanks.
Singular Value Decomposition for Continuous Variables
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linear-algebra
functional-analysis
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2Probably the more natural approach to the generalization would be to try to identify the spectral decomposition of $w^* w$ and of $w w^*$, where $w^*$ is the adjoint of $w$. Then the "eigenvector matrix" for the second one is your first factor, the "eigenvector matrix" for the first one is your third factor, and the "singular values matrix" should be straightforward to construct from the diagonal operator. – 2015-02-02
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1To do this, one would have to start with a suitable functionalanalytic framework, e.g. assume that $w$ is square integrable over something like $[a,b] \times [c,d]$. As suggested in the previous comment, set $w_1(x,z) = \int_c^d w(x,s)w(z,s) \, ds$ and $w_2(t,y) = \int_a^b w(s,y)w(s,t) ds$. These are symmetric and positive definite integral kernels and therefore can be written as $w_1(x,z) = \sum_{j=1}^\infty \alpha_j \phi_j(x) \phi_j(z), w_2(y,t) = \sum \alpha_j \psi_j(y) \psi_j(t)$. Then it should be possible to show that $w(x,y) = \sum \sqrt{\alpha_j} \phi_j(x) \psi_j(y)$. – 2015-02-08
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0Thank you both for your comments! I will look into this idea, as it's a natural extension from the matrix definition. – 2015-02-10