help with sum of rank-1 matrices

[ecoli]

Need help on a problem and I think there's some notation issues I'm having:

 

[math] A=\sum _{i}^{r}\lambda _{i}\bold{u_{i}v_{i}}^{T} [/math]

where A is a nxr matrix with (left and right) singular vectors v_1... v_r, u_1...u_r

& lambda are singular values

 

How does summation work in this case? Each product of rank-1 matrices results in a vector which, when summed, should result in a vector. But A is a matrix, so I feel like I must be missing something very simple about matrix addition.

 

[mississippichem]

The product "uv^T" gives an r x r matrix. You then have a sum of lamdas times that r x r matrix. Distribute the matrix over the sum of lamdas. I'll LaTeX the matrix out explicity if you neef further clarification.

Edited June 26, 2012 by mississippichem

[ecoli]

The product "uv^T" gives an r x r matrix. You then have a sum of lamdas times that r x r matrix. Distribute the matrix over the sum of lamdas. I'll LaTeX the matrix out explicity if you neef further clarification.

 

Ah now its obvious. I was performing this calculation in R... apparently, a single column/row subset of a matrix in R is treated as a vector, not a single-entry matrix. So the transpose operation failed. Hence the confusion.

 

Thanks!