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## Main Question or Discussion Point

Hi. I'm wondering if anyone can point me to any information on techniques to decompose a matrix (actually a 3D matrix) into an outer product of vectors. Particularly, given [tex]M_{i,j,k}[/tex], I want to find vectors [tex]a_{i}[/tex], [tex]b_{i}[/tex] and [tex]c_{i}[/tex] such that

[tex]

M_{i,j,k} = a_{i}b_{i}c_{i}

[/tex]

where the multiplication on the right is an outer product.

I've read that this is only possible if the matrix M has a rank of one, but I can find anything on how to actually decompose the matrix, only that it CAN BE done. Also, if M has rank one, does that mean that there is a "unique" decomposition? What if the rank is something other than one? In that case would it be possible to find a family of solutions?

Thanks for any help.

[tex]

M_{i,j,k} = a_{i}b_{i}c_{i}

[/tex]

where the multiplication on the right is an outer product.

I've read that this is only possible if the matrix M has a rank of one, but I can find anything on how to actually decompose the matrix, only that it CAN BE done. Also, if M has rank one, does that mean that there is a "unique" decomposition? What if the rank is something other than one? In that case would it be possible to find a family of solutions?

Thanks for any help.