Why Covariance Stays Positive

For any real matrix $X$ (data matrix, rows = samples), the matrix $X^\top X$ is always symmetric positive semi-definite. Why?

Because covariance matrices are always diagonal

Because eigenvalues of X are always real

For any vector z, $z^\top(X^\top X)z = |Xz|^2$ which is always $\geq 0$

Because X itself is always positive semi-definite

Because $X^\top X$ always has determinant 1