Symmetric matrix

Under a symmetric matrix is understood in linear algebra, a branch of mathematics, a matrix that is symmetric to the main diagonal.

Symmetric matrices are used to describe symmetric bilinear forms. In particular, real scalar are described by real symmetric matrices. In the theory of finite-dimensional real Prähilberträume symmetric matrices are used to describe self-adjoint Pictures: The matrix representation of a self-adjoint with respect to an orthonormal basis figure is symmetrical. In contrast, complex scalar and self-adjoint mappings are described in complex Prähilberträumen by Hermitian matrices.

Definition

A square matrix is called symmetric if it coincides with its transpose:

In other words, the matrix is symmetric if for their entries applies: for all.

Examples

Properties

Matrices over arbitrary fields

The set of symmetric matrices over the field is a vector subspace of the dimension.

Real matrices

Symmetric matrices on are normal. A real symmetric matrix is a normal matrix with a full set of eigenvectors, where the eigenvectors corresponding to different eigenvalues ​​are orthogonal diagonalizable.

In other words, is and is denoted by the orthonormal eigenvectors of, so goes with that or with a diagonal matrix having the eigenvalues ​​to the eigenvectors in the corresponding order on the diagonal.

From the property also follows immediately that is self-adjoint on a real vector space, and can therefore occur only real eigenvalues ​​.

Solution of linear systems of equations

Finding the solution of a linear system of equations with symmetric positive coefficient matrix is simplified if one takes advantage of the symmetry of the coefficient matrix. Due to the symmetry of the coefficient matrix can be written as a product with strict left lower triangular matrix and diagonal matrix. This decomposition is used, for example in the Cholesky decomposition to calculate the solution of the equation system.