why does R choose the negative one from multiple available solutions?

[shiqangpan]

here is a matrix

A = matrix(c(1,1,2,2,0,0), nrow = 3,byrow = TRUE)

this code is to find the svd for it

>svd(A)
$d
[1] 3.162278e+00 1.570092e-16

$u
           [,1]       [,2]
[1,] -0.4472136 -0.8944272
[2,] -0.8944272  0.4472136
[3,]  0.0000000  0.0000000

$v
           [,1]       [,2]
[1,] -0.7071068 -0.7071068
[2,] -0.7071068  0.7071068

for matrix A, both

$u
           [,1]       [,2]
[1,] -0.4472136 -0.8944272
[2,] -0.8944272  0.4472136
[3,]  0.0000000  0.0000000

and

$u
           [,1]       [,2]
[1,] 0.4472136  0.8944272
[2,] 0.8944272 -0.4472136
[3,] 0.0000000  0.0000000

are correct solution.

the question is: why does R choose the negative one?

for another case, R choose the positive one.

A = matrix(c(5,5,-1,7), nrow = 2,byrow = TRUE)
$u
          [,1]       [,2]
[1,] 0.7071068  0.7071068
[2,] 0.7071068 -0.7071068

[Yarnabrina]

The question is not clear to me.

@shiqangpan

why does R choose the negative one?

What do you mean by this? How do you tell whether a matrix or a vector is positive or negative?

This happens because of the calculation of the eigenvectors, and that can lead to results with varying signs. In any case, multiplying by (-1) has no effect. What matters is the direction of the eigenvector, and that doesn't get altered if scaled by (-1).

You can find these relevant:

1. Singular Value Decomposition (Check this section: Singular Value Decomposition Step-by-Step)

2. Sign of eigenvectors change depending on specification of the symmetric argument for symmetric matrices

[shiqangpan]

thanks for your reply, I know the procedure of SVD and I said

@shiqangpan

both ... are correct solution.

I just want to know the logic of putting negative or positive sign on the eigenvectors