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Concept
Symmetric matrix
Summary
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if denotes the entry in the th row and th column then
for all indices and
Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different from 2, 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 represented in an orthonormal basis 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 often 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 following matrix is symmetric:
Since .
The sum and difference of two symmetric matrices is symmetric.
This is not always true for the product: given symmetric matrices and , then is symmetric if and only if and commute, i.e., if .
For any integer , is symmetric if is symmetric.
If exists, it is symmetric if and only if is symmetric.
Rank of a symmetric matrix is equal to the number of non-zero eigenvalues of .
Any square matrix can uniquely be written as sum of a symmetric and a skew-symmetric matrix. This decomposition is known as the Toeplitz decomposition. Let denote the space of matrices. If denotes the space of symmetric matrices and the space of skew-symmetric matrices then and , i.e.
where denotes the direct sum. Let then
Notice that and . This is true for every square matrix with entries from any field whose characteristic is different from 2.
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