The eigen values of a symmetric matrix are all A. complex with non-zero positive imaginary part B. complex with non-zero negative imaginary part C. real D. pure imaginary

The correct answer is C. real.

A symmetric matrix is a square matrix that is equal to its transpose. The eigenvalues of a symmetric matrix are all real numbers. This is because the eigenvalues of a matrix are the roots of its characteristic polynomial, and the characteristic polynomial of a symmetric matrix is a real polynomial.

Option A is incorrect because the eigenvalues of a symmetric matrix are not necessarily complex. Option B is incorrect because the eigenvalues of a symmetric matrix are not necessarily complex with non-zero negative imaginary part. Option D is incorrect because the eigenvalues of a symmetric matrix are not necessarily pure imaginary.