In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Let A be a symmetric matrix. Then:
The entries of a symmetric matrix are symmetric with respect to the main diagonal (top left to bottom right). So if the entries are written as A = (aij), then
for all indices i and j. The following 3×3 matrix is symmetric:
Every diagonal matrix is symmetric, since all off-diagonal entries are zero. Similarly, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative.
In linear algebra, a real symmetric matrix represents a self-adjoint operator over a real inner product space. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose. Therefore, in linear algebra over the complex numbers, it is generally assumed that a symmetric matrix refers to one which has real-valued entries. Symmetric matrices appear naturally in a variety of applications, and typical numerical linear algebra software makes special accommodations for them.
The finite-dimensional spectral theorem says that any symmetric matrix whose entries are real can be diagonalized by an orthogonal matrix. More explicitly: For every symmetric real matrix A there exists a real orthogonal matrix Q such that D = QTAQ is a diagonal matrix. Every symmetric matrix is thus, up to choice of an orthonormal basis, a diagonal matrix.
Another way of stating the real spectral theorem is that the eigenvectors of a symmetric matrix are orthogonal. More precisely, a matrix is symmetric if and only if it has an orthonormal basis of eigenvectors.
Every real symmetric matrix is Hermitian, and therefore all its eigenvalues are real. (In fact, the eigenvalues are the entries in the above diagonal matrix D, and therefore D is uniquely determined by A up to the order of its entries.) Essentially, the property of being symmetric for real matrices corresponds to the property of being Hermitian for complex matrices.
The sum and difference of two symmetric matrices is again symmetric, but this is not always true for the product: given symmetric matrices A and B, then AB is symmetric if and only if A and B commute, i.e., if AB = BA. So for integer n, An is symmetric if A is symmetric. Two real symmetric matrices commute if and only if they have the same eigenspaces.
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