# .

# Symmetric matrix

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Let A be a symmetric matrix. Then:

\( A = A^{\top}. \,\! \)

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:

\( \begin{bmatrix} 1 & 7 & 3\\ 7 & 4 & -5\\ 3 & -5 & 6\end{bmatrix}. \)

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.

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 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.

Another way to phrase the spectral theorem is that a real n×n matrix A is symmetric if and only if there is an orthonormal basis of \( \mathbb{R}^n \) consisting of eigenvectors for A.

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.

A complex symmetric matrix *A* can often, but not always, be diagonalized in the form *D* = *U ^{T} A U*, where

*D*is complex diagonal and

*U*is not Hermitian but complex orthogonal with

*U*. In this case the columns of

^{T}U = I*U*are the eigenvectors of

*A*and the diagonal elements of

*D*are eigenvalues. An example of a complex symmetric matrix that cannot be diagonalized is

\( \begin{bmatrix} i& 1 \\ 1& -i\end{bmatrix}. \)

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 same eigenspaces.

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.

\( \mbox{Mat}_n = \mbox{Sym}_n \oplus \mbox{Skew}_n , \)

where ⊕ denotes the direct sum. Let X ∈ Matn then

\( X = \frac{1}{2}(X + X^{\top}) + \frac{1}{2}(X - X^{\top}) . \)

Notice that ½(X + XT) ∈ Symn and ½(X − XT) ∈ Skewn. 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*.

Denote with \( \langle \cdot,\cdot \rangle \) the standard inner product on Rn. The real n-by-n matrix A is symmetric if and only if

\( \langle Ax,y \rangle = \langle x, Ay\rangle \quad \forall x,y\in\Bbb{R}^n. \)

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.

A symmetric matrix is a normal matrix.

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.[1]

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, \( A=L L^T \). If the matrix is symmetric indefinite, it may be still decomposed as \( P A P^T = L T L^T \) where P is a permutation matrix (arising from the need to pivot), L a lower unit triangular matrix, T a symmetric tridiagonal matrix, and D a direct sum of symmetric 1×1 and 2×2 blocks.[2]

Every real symmetric matrix A can be diagonalized, moreover the eigen decomposition takes a simpler form:

\( A = Q \Lambda Q^{\top} \)

where Q is an orthogonal matrix (the columns of which are eigenvectors of A), and Λ is real and diagonal (having the eigenvalues of A on the diagonal).

Hessian

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 λi. 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}(D*

^{T}

*SD*). A matrix

*A*= (

*a*) is symmetrizable if and only if the following conditions are met:

_{ij} \( a_{ij} = 0 \text{ implies } a_{ji} = 0 \text{ for all } 1 \le i \le j \le n.

a_{i_1i_2} a_{i_2i_3}\dots a_{i_ki_1} = a_{i_2i_1} a_{i_3i_2}\dots a_{i_1i_k} \text{ for any finite sequence } (i_1, i_2, \dots, i_k). \)

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.

References

^ Bosch, A. J. (1986). "The factorization of a square matrix into two symmetric matrices". American Mathematical Monthly 93 (6): 462–464. doi:10.2307/2323471. JSTOR 2323471.

^ G.H. Golub, C.F. van Loan. (1996). Matrix Computations. The Johns Hopkins University Press, Baltimore, London.

External links

A brief introduction and proof of eigenvalue properties of the real symmetric matrix

Retrieved from "http://en.wikipedia.org/"

All text is available under the terms of the GNU Free Documentation License