A symmetric matrix is a square matrix that satisfies

where denotes the transpose, so . This also implies

where is the identity matrix. For example,

is a symmetric matrix. Hermitian matrices are a useful generalization of symmetric matrices for complex matrices.

A matrix that is not symmetric is said to be an asymmetric matrix, not to be confused with an antisymmetric matrix.

A matrix can be tested to see if it is symmetric in the Wolfram Language using SymmetricMatrixQ[m].

Written explicitly, the elements of a symmetric matrix have the form

The symmetric part of any matrix may be obtained from

A matrix is symmetric if it can be expressed in the form

where is an orthogonal matrix and is a diagonal matrix. This is equivalent to the matrix equation

which is equivalent to

for all , where . Therefore, the diagonal elements of are the eigenvalues of , and the columns of are the corresponding eigenvectors.

The numbers of symmetric matrices of order on symbols are , , , , ..., . Therefore, for (0,1)-matrices, the numbers of distinct symmetric matrices of orders , 2, ... are 2, 8, 64, 1024, ... (OEIS A006125).

See also

Antihermitian Matrix, Antisymmetric Matrix, Asymmetric Matrix, Bisymmetric Matrix, Conjugate Transpose, Hankel Matrix, Hermitian Matrix, Orthogonal Matrix, Symmetric Part

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References

Ayres, F. Jr. Schaum's Outline of Theory and Problems of Matrices. New York: Schaum, pp. 12 and 115-117, 1962.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.Sloane, N. J. A. Sequence A006125/M1897 in "The On-Line Encyclopedia of Integer Sequences."

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Symmetric Matrix

Cite this as:

Weisstein, Eric W. "Symmetric Matrix." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SymmetricMatrix.html

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