For a given data matrix $A$, it seems like $A^TA$ plays an important role in statistics. For example, it is an important part of the analytical solution of ordinary least squares. Or, for PCA, its eigenvectors are the principal components of the data.

I understand how to calculate $A^TA$, but I was wondering if there's an intuitive interpretation of what this matrix represents, which leads to its important role?

For a given data matrix $A$ (with variables in columns and data points in rows), it seems like $A^TA$ plays an important role in statistics. For example, it is an important part of the analytical solution of ordinary least squares. Or, for PCA, its eigenvectors are the principal components of the data.

I understand how to calculate $A^TA$, but I was wondering if there's an intuitive interpretation of what this matrix represents, which leads to its important role?