Symmetric Matrices

A symmetric matrix — equal to its own transpose — is the nicest matrix you'll ever meet. Its eigenvalues are always real (never complex), its eigenvectors are mutually perpendicular, and it diagonalizes with an orthogonal matrix. That bundle of good behavior is the spectral theorem, and it's why symmetric matrices run physics (energy forms), statistics (covariance), and machine learning (PCA). When a matrix is symmetric, the hard parts of linear algebra melt away.