When applying Principal Component Analysis to a high-dimensional dataset, how is the total variance of the original data preserved across the resulting principal components?

In Principal Component Analysis, the total variance of the original data is preserved because the transformation is a rigid rotation of the coordinate system that does not change the internal distances or the total spread of the data points. Before performing the analysis, the dataset is centered by subtracting the mean from each feature, ensuring the data is located at the origin. Variance is defined as the sum of the squared distances of all data points from this mean.....