I'm learning PCA in R language. I met two problems right now that I don't understand.
I am performing a PCA analysis in R on a 318×17 dataset using some custom code. I take eigen function in R to find eigenvalues and eigenvectors. But my 1st and 3rd eigenvectors are of the opposite sign to my handbook. My second eigenvectors is almost the same.
I know that given a square matrix A, the condition that characterizes an eigenvalue, $lambda$, is the existence of a nonzero vector $x$ such that $Ax=lambda x$; this equation can be rewritten as follows: $(A – lambda)x=0$.
Now I calculate covariance of my data and have eigenvalues. I want to solve this linear combination equation to find $x$ and compare with initial eigenvectors. When I take solve function in R, my $x$ vector is always zero.
Here are my questions: Why the sign is different? How to use solve function in R to find a non-zero vector $x$?
Best Answer
1) The definition of eigenvector $Ax = lambda x$ is ambidextrous. If $x$ is an eigenvector, so is $-x$, for then
$$A(-x) = -Ax = -lambda x = lambda (-x)$$
So the definition of an eigenbasis is ambiguous of sign.
2) It's hard to know for sure, but I have a strong suspicion of what is happening here. Your equation
$$ (A – lambda)x = 0 $$
is technically incorrect. The correct equation is
$$ (A – lambda I)x$$
The first equation is often used as a shorthand for the second. In general, this is unambiguous, because there is no real mathematical way to subtract a vector from a square matrix, but it is abuse of notation. In R though, you have broadcasting. So if you do
> M <- matrix(c(1, 1, 1, 1), nrow=2) > M - .5 [,1] [,2] [1,] 0.5 0.5 [2,] 0.5 0.5 its not really what you want. The proper way would be
> M - diag(.5, 2) [,1] [,2] [1,] 0.5 1.0 [2,] 1.0 0.5 The reason you are getting zero solutions is that the matrix you are starting with $A$ is invertible. More than likely (almost surely), the matrix you get by subtracting the same number from every entry will also be invertible. For invertible matrices, the only solution to $Ax = 0$ is the zero vector.
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