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:A^ extrm{T} = A

which implies that ''A'' is a Square Matrix . 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-by-3 matrix is symmetric:

:\begin{bmatrix}
1 & 2 & 3\
2 & -4 & 5\
3 & 5 & 6\end{bmatrix}

Any Diagonal Matrix is symmetric, since all its off-diagonal entries are zero.

A matrix is called Skew-symmetric if its transpose is the same as its negative.


PROPERTIES


One of the basic theorems concerning such matrices is the finite-dimensional choice of an Orthonormal Basis , a diagonal matrix.

Another way of stating the spectral theorem is that the Eigenvector s of a symmetric matrix are orthogonal.

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 symmetry of 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= rac{1}{2}\left(X+X^ extrm{T} ight)+ rac{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 , i.e. if ''AB'' = ''BA''. Two real symmetric matrices commute if and only if they have the same Eigenspace s.

Denote with <,> 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,y\in\Bbb{R}^n.

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)


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 of Conic 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


Other types of Symmetry or pattern in square matrices have special names; see for example:


See also Symmetry In Mathematics .


REFERENCES