- Symmetric matrix
In

linear algebra , a**symmetric matrix**is asquare matrix , "A", that is equal to itstranspose :$A\; =\; A^\{T\}.\; ,!$

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" = ("a"_{"ij"}), then:$a\_\{ij\}\; =\; a\_\{ji\}\; ,!$for all indices "i" and "j". The following 3×3 matrix is symmetric::$egin\{bmatrix\}1\; 2\; 3\backslash 2\; 4\; -5\backslash 3\; -5\; 6end\{bmatrix\}.$

A matrix is called skew-symmetric or antisymmetric if its transpose is the same as its negative. The following 3×3 matrix is skew-symmetric:

:$egin\{bmatrix\}0\; -3\; 4\backslash 3\; 0\; -5\backslash -4\; 5\; 0end\{bmatrix\}.$

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. The following matrix is neither symmetric nor skew-symmetric::$egin\{bmatrix\}1\; -4\; 2\backslash 5\; 1\; -4\backslash -3\; 5\; 1end\{bmatrix\}.$

In linear algebra, a symmetric matrix represents a

self-adjoint operator over a realinner product space . The corresponding object for a complex inner product space is aHermitian matrix with complex-valued entries, which is equal to itsconjugate transpose . Therefore, it is generally assumed that a symmetric matrix 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**One of the basic theorems concerning such matrices is the finite-dimensional

spectral theorem , which says that any symmetric matrix whose entries are real can be diagonalized by anorthogonal 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 anorthonormal basis , a diagonal matrix.Another way of stating the real spectral theorem is that the

eigenvector s 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.

Every square real matrix "X" can be written in a unique way as the sum of a symmetric and a

skew-symmetric matrix. This is done in the following way::$X=frac\{1\}\{2\}left(X+X^\; extrm\{T\}\; ight)+frac\{1\}\{2\}left(X-X^\; extrm\{T\}\; ight).$(This is true more generally for every square matrix "X" with entries from any field whose characteristic is different from 2.)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. Two real symmetric matrices commute if and only if they have the sameeigenspace s.If "A"

^{−1}exists, it is symmetric if "A" is symmetric.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".Denote with $langle\; cdot,cdot\; angle$ the standard

inner product on**R**^{"n"}. The real "n"-by-"n" matrix "A" is symmetric if and only if :$langle\; Ax,y\; angle\; =\; langle\; x,\; Ay\; angle\; quad\; mbox\{for\; all\; \}x,yinBbb\{R\}^n.$It should be noted that this definition is independent of the choice of

basis , and thus symmetry is a property that depends only on thelinear operator A and a choice ofinner product . In finite dimensions, the relationship between linear maps or operators and matrices is so close, that one often speaks of them almost interchangeably. But this basis independent definition of symmetry is often important. For example, indifferential geometry eachtangent space to amanifold may be endowed with an inner product, giving rise to what is called aRiemannian manifold . It may be convenient to work with explicit coordinates, but often it is not. If we do not wish to do so, we may require thattangent space s be endowed with a non-degenerate symmetric form, and when a basis is fixed, this reduces to the familiar case of a symmetric matrix. Another area where this formulation is important is in infinite dimensional spaces calledHilbert space s where it is simply not possible to write down a matix representation.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. (Bosch, 1986)Every real

non-singular matrix can be uniquely factored as the product of anorthogonal matrix and a symmetricpositive definite matrix , which is called apolar decomposition . Singular matrices can be also factored, but not uniquely.A symmetric $n\; imes\; n$ matrix is determined by $frac\{n(n+1)\}\{2\}$ scalars. Similarly, a skew-symmetric matrix is determined by $frac\{n(n-1)\}\{2\}$ scalars.

**Occurrence**Symmetric real "n"-by-"n" matrices appear as the Hessian 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":$q(x\_1,ldots,x\_n)=sum\_\{i=1\}^n\; lambda\_i\; x\_i^2$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 ofconic section s.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 .**See also***

Hermitian matrix

*Normal matrix

*Hilbert matrix

*Self-adjoint operator Other types of

symmetry or pattern in square matrices have special names; see for example:*

Circulant matrix

*Hankel matrix

*Toeplitz matrix

*Centrosymmetric matrix See also

symmetry in mathematics .**References***

**External links*** [

*http://www.ocolon.org/editor/template.php?.matrix_split_symm_skew Template for splitting a matrix online into a symmetric and a skew-symmetric addend*]

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