Question-4

PCA is performed on a mean-centered dataset in \(\mathbb{R}^{3}\). If the first principal component is \(\cfrac{1}{\sqrt{6}}(1, -1, 2)\), which of the following could be the second principal component?

• \((0, 0, 0)\)

• \(\cfrac{1}{\sqrt{2}}(-1, -1, 0)\)

• \((0, 1, 2)\)

• \(\cfrac{1}{\sqrt{5}}(0, 1, 2)\)

Answer

• \((0, 0, 0)\)

• \(\cfrac{1}{\sqrt{2}}(-1, -1, 0)\)

• \((0, 1, 2)\)

• \(\cfrac{1}{\sqrt{5}}(0, 1, 2)\)

Solution

The principal components are orthogonal eigenvectors of the covariance matrix. Since they are eigenvectors, \((0, 0, 0)\) is ruled out. Since they have to be orthogonal, only the second option is correct.