Symmetric matrix - Wikipedia

Abstraction: Square matrix equal to its transpose with real eigenvalues

Key points:

• A symmetric matrix satisfies A = A^T; every square matrix decomposes uniquely into symmetric and skew-symmetric parts (Toeplitz decomposition)
• Every real symmetric matrix is diagonalizable by an orthogonal matrix (spectral theorem): A = Q D Q^T with D diagonal and Q orthogonal
• All eigenvalues of a real symmetric matrix are real; rank equals the number of non-zero eigenvalues
• Two commuting real symmetric matrices can be simultaneously diagonalized by a single orthogonal matrix
• Complex symmetric matrices admit Autonne-Takagi factorization (unitary diagonalization with non-negative real diagonal), distinct from the Hermitian case
• Every quadratic form can be written as x^T A x for a unique symmetric matrix A, and by the spectral theorem looks like a sum of squares in an orthonormal basis
• Real positive-definite symmetric matrices admit Cholesky decomposition: A = L L^T

Connections: Linear Algebra · Spectral Theorem · Matrix Decomposition

Source: http://en.wikipedia.org/wiki/Symmetric_matrix