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The correct answer is False.<\/p>\n
The principal components are the eigenvectors of the covariance matrix of the data, while the left singular values are the singular values of the data matrix. The covariance matrix is not affected by scaling the variables, but the data matrix is. Therefore, the principal components are not equal to the left singular values if you first scale the variables.<\/p>\n
To elaborate, the covariance matrix of a set of data is a square matrix that contains the covariances between each pair of variables. The covariance between two variables is a measure of how much they vary together. The covariance matrix is symmetric, which means that the covariance between $x_i$ and $x_j$ is equal to the covariance between $x_j$ and $x_i$.<\/p>\n
The singular value decomposition (SVD) of a matrix is a way of decomposing it into a product of three matrices. The first matrix is a unitary matrix, which means that it has orthonormal columns. The second matrix is a diagonal matrix, which means that all of its entries are either 0 or 1. The third matrix is also a unitary matrix.<\/p>\n
The left singular values of a matrix are the diagonal entries of the diagonal matrix in the SVD of the matrix. The principal components of a set of data are the eigenvectors of the covariance matrix of the data. The eigenvectors of a matrix are the columns of the unitary matrix in the SVD of the matrix.<\/p>\n
Therefore, the principal components are not equal to the left singular values if you first scale the variables. This is because the covariance matrix is not affected by scaling the variables, but the data matrix is.<\/p>\n","protected":false},"excerpt":{"rendered":"
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