I'm learning Linear Algebra with Applications 8th Edition by Steve J Leon. The Theorem about my question is the following:
Theorem 6.5.1 The SVD Theorem
If $A$ is an $m × n$ matrix, then $A$ has a singular value decomposition.
The proof given is the following:
Proof
$A^TA$ is a symmetric $n × n$ matrix. Therefore, its eigenvalues are all real and it has an orthogonal diagonalizing matrix ...
Why are eigenvalues of $A^TA$ all real here? I know that eigenvalues of a REAL symmetric matrix are all real, but I cannot find any assumption that A is a real matrix from context. Is it often assumed that the matrix will be decomposed is REAL for The Singular Value Decomposition? At the forefront of this section is "In many applications, ...", is that assumed the matrix is REAL?