# Symmetric matrix

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^{[1]} 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.

## Contents

## 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*, *A ^{n}* 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 *n* × *n* 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 *n* × *n* symmetric matrices and Skew_{n} the space of *n* × *n* skew-symmetric matrices then since Mat_{n} = Sym_{n} + Skew_{n} and Sym_{n} ∩ Skew_{n} = {0}, i.e.

where ⊕ denotes the direct sum. Let X ∈ Mat_{n} then

Notice that 1/2(*X* + *X*^{T}) ∈ Sym_{n} and 1/2(*X* − *X*^{T}) ∈ Skew_{n}. 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.

### Complex symmetric matrices

A complex symmetric matrix can be diagonalized using a unitary matrix: thus if *A* is a complex symmetric matrix, there is a unitary matrix *U* such that
*UAU*^{t} is a diagonal matrix. This result is referred to as the **Autonne–Takagi factorization**. It was originally proved by Leon Autonne (1915) and Teiji Takagi (1925) and rediscovered with different proofs by several other mathematicians.^{[2]}^{[3]} In fact the matrix *B* = *A***A* is Hermitian and non-negative, so there is a unitary matrix *V* such that *V***BV* is diagonal with non-negative real entries. Thus *C* = *V*^{t}*AV* is complex symmetric with *C***C* real. Writing *C* = *X* + *iY* with *X* and *Y* real symmetric matrices, *C***C* = *X*^{2} − *Y*^{2} + i
(*XY* − *YX*). Thus *XY* = *YX*. Since *X* and *Y* commute, there is a real orthogonal matrix *W* such that *WXW*^{t} and *WYW*^{t} are diagonal. Setting *U* = *WV*^{t}, the matrix *UAU*^{t} is diagonal. Post-multiplying *U* by a diagonal matrix the diagonal entries can be taken to be non-negative. Since their squares are the eigenvalues of *A***A*, they coincide with the singular values of *A*.

## 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.^{[4]}

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 *A* is a product of a lower-triangular matrix *L* and its transpose, .
If the matrix is symmetric indefinite, it may be still decomposed as where is
a permutation matrix (arising from the need to pivot), a lower unit triangular matrix, a symmetric tridiagonal matrix, and
a direct sum of symmetric 1×1 and 2×2 blocks.^{[5]}

A complex symmetric matrix need not be diagonalizable by similarity; every real symmetric matrix is diagonalizable by a real orthogonal similarity.

Every complex symmetric matrix *A* can be diagonalized by unitary congruence

where *Q* is an unitary matrix. If A is real *Q* is a real orthogonal matrix, (the columns of which are eigenvectors of *A*), and *Λ* is real and diagonal (having the eigenvalues of *A* on the diagonal). To see orthogonality, suppose and are eigenvectors corresponding to distinct eigenvalues , . Then

so that if then , a contradiction; hence .

## Hessian

Symmetric *n*-by-*n* matrices of real functions appear as the Hessians of twice continuously differentiable functions of *n* real variables.

Every quadratic form *q* on **R**^{n} can be uniquely written in the form *q*(**x**) = **x**^{T}*A***x** with a symmetric *n*-by-*n* matrix *A*. Because of the above spectral theorem, one can then say that every quadratic form, up to the choice of an orthonormal basis of **R**^{n}, "looks like"

with real numbers λ_{i}. This considerably simplifies the study of quadratic forms, as well as the study of the level sets {**x** : *q*(**x**) = 1} 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 *n*-by-*n* matrix *A* is said to be **symmetrizable** if there exist an invertible diagonal matrix *D* and symmetric matrix *S* such that *A* = *DS*.
The transpose of a symmetrizable matrix is symmetrizable, for (*DS*)^{T} = *D ^{−T}(*DSD

*)*

^{T}.*A matrix*A

*= (*a

_{ij}

*) 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:

- Antimetric matrix
- Centrosymmetric matrix
- Circulant matrix
- Covariance matrix
- Coxeter matrix
- Hankel matrix
- Hilbert matrix
- Persymmetric matrix
- Skew-symmetric matrix
- Toeplitz matrix

See also symmetry in mathematics.

## Notes

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## References

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## External links

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