# Multiplying an eigenvector by $-1$ while constructing the V matrix in SVD decomposition

While performing SVD I found eigenvectors, which will allow me to write down my $$V$$ matrix. However, I need to multiply the 3rd eigenvector by $$-1$$ because it will satisfy some condition that my task contains.

I understand that multiplying by $$-1$$ won't affect the ortogonality of all the eigenvectors, because my new eigenvector will just face the opposite direction. Nor will it affect normality, as the vector magnitude won't be affected either.

But I'm not sure about the matrices $$V$$ and $$U$$.

Question: Do I have to multiply their corresponding columns by $$-1$$ as well?