In linear algebra and bilinear, a symmetrical matrix is a square matrix which equal to its characteristic is transposed.

because P is a symmetric positive-definite matrix.

Note that the covariance matrix is always a positive semi-definite matrix.

Note that $A^TA$ is a positive definite symplectic matrix.

The linear coefficients (Lij) of an equation system derived from the linear combinations form a symmetrical matrix.

Example: A diagonal matrix and its associated quadratic form.

A real symmetric $n \times n$ matrix $A$ is called positive definite if

All eigenvalues of a positive definite matrix are positive.

All positive integral powers of symmetric matrix is symmetric.

Inverse of the Hilbert matrix of size n. c is a real, symmetric matrix.

Express a matrix $A$ as the sum of a symmetric and a skew symmetric matrix

Then the principal curvatures are the eigenvalues of the symmetric matrix

Since is a symmetric real matrix, it can be written in the form:

Since the middle matrix is symmetric, it can be diagonalized.

The 3 x 3 symmetric matrix is called the effective reciprocal inertia tensor.