Hermitian matrices (including symmetric matrices) have real eigenvalues and orthonormal eigenvectors. The spectral theorem guarantees a complete set of eigenvectors. Symmetric positive definite matrices have positive eigenvalues. This is crucial for many applications, especially those involving quadratic energies. This segment explains the spectral theorem, a crucial result stating that Hermitian matrices (including symmetric matrices) possess orthonormal eigenvectors forming a complete set. The speaker connects this theorem to the real eigenvalues of symmetric matrices and their significance in quadratic energy contexts, highlighting the practical implications of this theoretical result for various applications involving symmetric positive definite matrices which frequently arise in quadratic energy problems.