Test if a real machine-precision matrix is symmetric:

A real symmetric matrix is also Hermitian:

Test if a complex matrix is symmetric:

A complex symmetric matrix has symmetric real and imaginary parts:

Test if an exact matrix is symmetric:

Make the matrix symmetric:

Use SymmetricMatrixQ with an arbitrary-precision matrix:

A random matrix is typically not symmetric:

Use SymmetricMatrixQ with a symbolic matrix:

The matrix becomes symmetric when :

SymmetricMatrixQ works efficiently with large numerical matrices:

Use SymmetricMatrixQ with sparse matrices:

Use SymmetricMatrixQ with structured matrices:

Use with a QuantityArray structured matrix:

The identity matrix is symmetric:

HilbertMatrix is symmetric:

This matrix is symmetric for a positive real , but SymmetricMatrixQ gives False :

Use the option SameTest to get the correct answer:

Generate a real-valued symmetric matrix with some random perturbation of order 10^-14:

Adjust the option Tolerance to accept this matrix as symmetric:

The norm of the difference between the matrix and its transpose:

Any matrix generated from a symmetric function is symmetric:

The function is symmetric:

Using Table generates a symmetric matrix:

SymmetrizedArray can generate matrices (and general arrays) with symmetries:

Convert back to an ordinary matrix using Normal :

Check that matrices drawn from GaussianOrthogonalMatrixDistribution are symmetric:

Matrices drawn from CircularOrthogonalMatrixDistribution are symmetric and unitary:

Every Jordan matrix is similar to a symmetric matrix. Since any square matrix is similar to its Jordan form, this means that any square matrix is similar to a symmetric matrix. Define a function for generating an Jordan block for eigenvalue :

For example, here is the Jordan matrix of dimension 4 for the eigenvalue :

Define a function for generating a corresponding complex similarity transformation:

The matrix is a sum of times the identity matrix and times the backward identity matrix:

Then s(n).j(lambda,n).TemplateBox[{{s, (, n, )}}, Inverse] is symmetric, which shows that the Jordan matrix is similar to a symmetric matrix:

Confirm the matrix is symmetric:

The Hessian matrix of a function is symmetric:

Many special matrices are symmetric, including FourierMatrix :

HadamardMatrix :

HankelMatrix :

And HilbertMatrix :

Visualize the matrix types:

Many filter kernel matrices are symmetric, including DiskMatrix :

CrossMatrix :

DiamondMatrix :

Visualize the matrices:

AdjacencyMatrix of an undirected graph is symmetric:

As is KirchhoffMatrix :

Visualize adjacency and Kirchhoff matrices for different graphs:

Several statistical measures are symmetric matrices, including Covariance :

Correlation :

AbsoluteCorrelation :

A positive-definite, real symmetric matrix or metric defines an inner product by :

Verify that is in fact symmetric and positive definite:

Orthogonalize the standard basis of TemplateBox[{}, Reals]^n to find an orthonormal basis:

Confirm that this basis is orthonormal with respect to the inner product :

The moment of inertia tensor is the equivalent of mass for rotational motion. For example, kinetic energy is , with taking the place of the mass and angular velocity taking the place of linear velocity in the formula . can be represented by a positive-definite symmetric matrix. Compute the moment of inertia for a tetrahedron with endpoints at the origin and positive coordinate axes:

Verify that the matrix is symmetric:

Compute the kinetic energy if its angular velocity is :

The kinetic energy is positive as long as is nonzero, showing the matrix was positive definite:

Determine if a sparse matrix is structurally symmetric:

The matrix is not symmetric:

But it is structurally symmetric:

Use a different method for symmetric matrices, with failover to a general method:

Construct real-valued matrices for testing:

For a non-symmetric matrix m, the function myLS just uses Gaussian elimination:

For a symmetric indefinite matrix ms, try Cholesky and continue with Gaussian elimination:

For a symmetric positive-definite matrix mpd, try Cholesky, which succeeds: