# Interpreting product of left singular matrix and vector of singular values in singular value decomposition

Suppose I decompose a matrix A using singular value decomposition, and I obtain

$A = UDV^T$

where $D$ is a diagonal matrix with the singular values along the diagonal entries.

Suppose that $d$ is the vector of singular values. Does the vector

$w = Ud$

have any practical or geometric interpretation?

To provide some practical context, I am performing SVD on a document-term matrix, which records documents along the rows and words along the columns. Each entry is the number of occurrences of a word in the document.

• Rather than giving us a putative answer and asking what the associated question might be, why don't you explain the problem you actually have and ask a specific question about it? That would fit the format of this site much better. – whuber Mar 15 '17 at 19:53
• This IS the problem that I have. I inherited someone's script in my lab, and her script contains this calculation, but she didn't explain why she did it or what it means. I'm trying to decipher the script on my own. – MSE Mar 16 '17 at 15:03
• That certainly demonstrates relevance, although unfortunately it offers few clues. Do you have any sense of the objective of this part of the code? If not, perhaps observing that $Ud=UD\mathbf{1}$ (where $\mathbf{1}$ is a vector of ones) could help with the interpretation. – whuber Mar 16 '17 at 21:44
• $UD$ is the matrix holding the PC scores one would get from PCA. $Ud$ is a vector holding the sum of the PC scores related to each document. I think it is practically useless... Maybe it can be interpreted to hold a vague relation to outlier detection but even that seems quite dodgy to me. It will have mean zero and (assuming $A$ was centred and scaled before its SVD) variance equal to the number of word terms used in $A$ (if that is any consolation)... – usεr11852 Mar 26 '17 at 1:17