## Symmetric Matrix

A symmetric matrix is a Square Matrix which satisfies where denotes the Transpose, so . This also implies

 (1)

where I is the Identity Matrix. Written explicitly,
 (2)

The symmetric part of any Matrix may be obtained from
 (3)

A Matrix A is symmetric if it can be expressed in the form
 (4)

where is an Orthogonal Matrix and is a Diagonal Matrix. This is equivalent to the Matrix equation
 (5)

which is equivalent to
 (6)

for all , where . Therefore, the diagonal elements of are the Eigenvalues of , and the columns of are the corresponding Eigenvectors.

Nash, J. C. Real Symmetric Matrices.'' Ch. 10 in Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation, 2nd ed. Bristol, England: Adam Hilger, pp. 119-134, 1990.