Symmetric Matrix - Decomposition

Decomposition

Using the Jordan normal form, one can prove that every square real matrix can be written as a product of two real symmetric matrices, and every square complex matrix can be written as a product of two complex symmetric matrices.

Every real non-singular matrix can be uniquely factored as the product of an orthogonal matrix and a symmetric positive definite matrix, which is called a polar decomposition. Singular matrices can also be factored, but not uniquely.

Cholesky decomposition states that every real positive-definite symmetric matrix A is a product of a lower-triangular matrix L and its transpose, . If the matrix is symmetric indefinite, it may be still decomposed as where is a permutation matrix (arising from the need to pivot), a lower unit triangular matrix, a symmetric tridiagonal matrix, and a direct sum of symmetric 1×1 and 2×2 blocks.

Every real symmetric matrix A can be diagonalized, moreover the eigen decomposition takes a simpler form:

where Q is an orthogonal matrix (the columns of which are eigenvectors of A), and Λ is real and diagonal (having the eigenvalues of A on the diagonal).