Symmetric matrix

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 formula_1 denotes the entry in the formula_2-th row and formula_3-th column then
for all indices formula_2 and formula_5
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 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.
Example.
The following formula_6 matrix is symmetric:
Properties.
Decomposition into symmetric and skew-symmetric.
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 formula_19 denote the space of formula_20 matrices. If formula_21 denotes the space of formula_20 symmetric matrices and formula_23 the space of formula_20 skew-symmetric matrices then formula_25 and formula_26, i.e.
where formula_28 denotes the direct sum. Let formula_29 then
Notice that formula_31 and formula_32. This is true for every square matrix formula_33 with entries from any field whose characteristic is different from 2.
A symmetric formula_20 matrix is determined by formula_35 scalars (the number of entries on or above the main diagonal). Similarly, a skew-symmetric matrix is determined by formula_36 scalars (the number of entries above the main diagonal).
Matrix congruent to a symmetric matrix.
Any matrix congruent to a symmetric matrix is again symmetric: if formula_33 is a symmetric matrix then so is formula_38 for any matrix formula_8.
Symmetry implies normality.
A (real-valued) symmetric matrix is necessarily a normal matrix.
Real symmetric matrices.
Denote by formula_40 the standard inner product on formula_41. The real formula_20 matrix formula_8 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 formula_8 there exists a real orthogonal matrix formula_46 such that formula_47 is a diagonal matrix. Every symmetric matrix is thus, up to choice of an orthonormal basis, a diagonal matrix.
If formula_8 and formula_9 are formula_20 real symmetric matrices that commute, then they can be simultaneously diagonalized: there exists a basis of formula_41 such that every element of the basis is an eigenvector for both formula_8 and formula_9.
Every real symmetric matrix is Hermitian, and therefore all its eigenvalues are real. (In fact, the eigenvalues are the entries in the diagonal matrix formula_54 (above), and therefore formula_54 is uniquely determined by formula_8 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.
Complex symmetric matrices.
A complex symmetric matrix can be 'diagonalized' using a unitary matrix: thus if formula_8 is a complex symmetric matrix, there is a unitary matrix formula_58 such that formula_59 is a real diagonal matrix with non-negative entries. This result is referred to as the Autonne–Takagi factorization. It was originally proved by Léon Autonne (1915) and Teiji Takagi (1925) and rediscovered with different proofs by several other mathematicians. In fact, the matrix formula_60 is Hermitian and positive semi-definite, so there is a unitary matrix formula_61 such that formula_62 is diagonal with non-negative real entries. Thus formula_63 is complex symmetric with formula_64 real. Writing formula_65 with formula_33 and formula_67 real symmetric matrices, formula_68. Thus formula_69. Since formula_33 and formula_67 commute, there is a real orthogonal matrix formula_72 such that both formula_73 and formula_74 are diagonal. Setting formula_75 (a unitary matrix), the matrix formula_76 is complex diagonal. Pre-multiplying formula_58 by a suitable diagonal unitary matrix (which preserves unitarity of formula_58), the diagonal entries of formula_76 can be made to be real and non-negative as desired. To construct this matrix, we express the diagonal matrix as formula_80. The matrix we seek is simply given by formula_81. Clearly formula_82 as desired, so we make the modification formula_83. Since their squares are the eigenvalues of formula_84, they coincide with the singular values of formula_8. (Note, about the eigen-decomposition of a complex symmetric matrix formula_8, the Jordan normal form of formula_8 may not be diagonal, therefore formula_8 may not be diagonalized by any similarity transformation.)
Decomposition.
Using the Jordan normal form, one can prove that every square real matrix can be written as a product of two real symmetric matrices, and every square complex matrix can be written as a product of two complex symmetric matrices.
Every real non-singular matrix can be uniquely factored as the product of an orthogonal matrix and a symmetric positive definite matrix, which is called a polar decomposition. Singular matrices can also be factored, but not uniquely.
Cholesky decomposition states that every real positive-definite symmetric matrix formula_8 is a product of a lower-triangular matrix formula_90 and its transpose,
If the matrix is symmetric indefinite, it may be still decomposed as formula_92 where formula_93 is a permutation matrix (arising from the need to pivot), formula_90 a lower unit triangular matrix, and formula_54 is a direct sum of symmetric formula_96 and formula_97 blocks, which is called Bunch–Kaufman decomposition 
A general (complex) symmetric matrix may be defective and thus not be diagonalizable. If formula_8 is diagonalizable it may be decomposed as
where formula_46 is an orthogonal matrix formula_101, and formula_102 is a diagonal matrix of the eigenvalues of formula_8. In the special case that formula_8 is real symmetric, then formula_46 and formula_102 are also real. To see orthogonality, suppose formula_107 and formula_108 are eigenvectors corresponding to distinct eigenvalues formula_109, formula_110. Then
Since formula_109 and formula_110 are distinct, we have formula_114.
Hessian.
Symmetric formula_20 matrices of real functions appear as the Hessians of twice continuously differentiable functions of formula_14 real variables.
Every quadratic form formula_117 on formula_41 can be uniquely written in the form formula_119 with a symmetric formula_20 matrix formula_8. Because of the above spectral theorem, one can then say that every quadratic form, up to the choice of an orthonormal basis of formula_41, "looks like"
with real numbers formula_124. This considerably simplifies the study of quadratic forms, as well as the study of the level sets formula_125 which are generalizations of conic sections.
This is important partly because the second-order behavior of every smooth multi-variable function is described by the quadratic form belonging to the function's Hessian; this is a consequence of Taylor's theorem.
Symmetrizable matrix.
An formula_20 matrix formula_8 is said to be symmetrizable if there exists an invertible diagonal matrix formula_54 and symmetric matrix formula_129 such that formula_130
The transpose of a symmetrizable matrix is symmetrizable, since formula_131 and formula_132 is symmetric. A matrix formula_133 is symmetrizable if and only if the following conditions are met:
See also.
Other types of symmetry or pattern in square matrices have special names; see for example:
See also symmetry in mathematics.