My question is generally on Singular Value Decomposition (SVD), and particularly on  Latent Semantic Indexing (LSI).

Say, I have $ A_{word \times document} $ that contains frequencies of 5 words for 7 documents.

    A =  matrix(data=c(2,0,8,6,0,3,1,
                       1,6,0,1,7,0,1,
                       5,0,7,4,0,5,6,
                       7,0,8,5,0,8,5,
                       0,10,0,0,7,0,0), ncol=7, byrow=TRUE)
    rownames(A) <- c('doctor','car','nurse','hospital','wheel')

I get the matrix factorization for $A$ by using SVD: $A = U \cdot D \cdot V^T $. 

    s = svd(A)
    D = diag(s$d) # singular value matrix
S = diag(s$d^0.5 ) # diag matrix with square roots of singular values.

In [1][1] and [2][2], it is stated that: 

$WordSim = U \cdot S$ gives the  **word similarity matrix**, where the rows of $WordSim $ represent different words.  

`WordSim =  s$u %*% S`


$DocSim= S \cdot V^T$ gives the  **document similarity matrix** where the columns of $DocSim$ represent different documents.

`DocSim = S %*% t(s$v)`

**Questions:**

1. Algebraically, why are $WordSim$ and $DocSimS$ word/document similarity matrices? Is there an intuitive explanation?
2. Based on the R example given, can we make any intuitive word count / similarity observations by just looking at $WordSim$ and $DocSim$ (without using cosine similarity or correlation coefficient between rows / columns)? 

![enter image description here][3]


  [1]: http://www.ling.ohio-state.edu/~kbaker/pubs/Singular_Value_Decomposition_Tutorial.pdf
  [2]: http://files.grouplens.org/papers/webKDD00.pdf
  [3]: https://i.sstatic.net/3KJuT.png