It is proved that if a is a matrix of order s * n, then the eigenvalues of ATA are all nonnegative real numbers

It is proved that if a is a matrix of order s * n, then the eigenvalues of ATA are all nonnegative real numbers

(the conclusion is only limited to the range of real numbers, and the transpose of complex numbers needs to be changed to conjugate transpose.)
Since ATA is a symmetric matrix ((ATA) t = ATA)), and symmetric matrix is semi positive definite if and only if its eigenvalues are non negative real numbers, so we only need to prove that this matrix is semi positive definite. Then any n-dimensional vector x, XT (ATA) x = (xtat) (AX) = (AX) t (AX), which is the inner product of (AX) and itself, obviously it is non negative, which = 0 if and only if AX = 0, indicates that ATA is semi positive definite, So its eigenvalues are all nonnegative real numbers