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In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, matrix ''A'' is symmetric if : 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 the entries are written as ''A'' = (''a''''ij''), then ''a''''ij'' = a''ji'', for all indices ''i'' and ''j''. The following 3×3 matrix is symmetric: : Every square 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 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. == Properties == 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. If ''A''−1 exists, it is symmetric if and only if ''A'' is symmetric. Let Mat''n'' denote the space of matrices. A symmetric ''n'' × ''n'' matrix is determined by ''n''(''n'' + 1)/2 scalars (the number of entries on or above the main diagonal). Similarly, a skew-symmetric matrix is determined by ''n''(''n'' − 1)/2 scalars (the number of entries above the main diagonal). If Sym''n'' denotes the space of symmetric matrices and Skew''n'' 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 ''X'' with entries from any field whose characteristic is different from 2. Any matrix congruent to a symmetric matrix is again symmetric: if ''X'' is a symmetric matrix then so is ''AXA''T for any matrix ''A''. A symmetric matrix is necessarily a normal matrix. === Real symmetric matrices === Denote by the standard inner product on R''n''. The real ''n''-by-''n'' matrix ''A'' is symmetric if and only if : Since this definition is independent of the choice of basis, symmetry is a property that depends only on the linear operator A and a choice of inner product. This characterization of symmetry is useful, for example, in differential geometry, for each tangent space to a manifold may be endowed with an inner product, giving rise to what is called a Riemannian manifold. Another area where this formulation is used is in Hilbert spaces. 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'' = ''Q''T''AQ'' is a diagonal matrix. Every symmetric matrix is thus, up to choice of an orthonormal basis, a diagonal matrix. If ''A'' and ''B'' are ''n''×''n'' real symmetric matrices that commute, then they can be simultaneously diagonalized: there exists a basis of such that every element of the basis is an eigenvector for both ''A'' and ''B''. Every real symmetric matrix is Hermitian, and therefore all its eigenvalues are real. (In fact, the eigenvalues are the entries in the diagonal matrix ''D'' (above), 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「symmetric matrix」の詳細全文を読む スポンサード リンク
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