Definitions

symmetric matrices

Symmetric matrix

In linear algebra, a symmetric matrix is a square matrix, A, that is equal to its transpose

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 = (aij), then

a_{ij} = a_{ji} ,!
for all indices i and j. The following 3×3 matrix is symmetric:

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

begin{bmatrix}
0 & -3 & 4 3 & 0 & -5 -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:

begin{bmatrix}
1 & -4 & 2 5 & 1 & -4 -3 & 5 & 1end{bmatrix}.

In linear algebra, a 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, 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 an orthogonal matrix. More explicitly: For every symmetric real matrix A there exists a real orthogonal matrix Q such that D = QTAQ is a diagonal matrix. Every symmetric matrix is thus, up to choice of an orthonormal basis, a diagonal matrix.

Another way of stating the real spectral theorem is that the eigenvectors 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^textrm{T}right)+frac{1}{2}left(X-X^textrm{T}right).
(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, An 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 A is symmetric.

Any matrix congruent to a symmetric matrix is again symmetric: if X is a symmetric matrix then so is AXAT 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, Ayrangle 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 the linear operator A and a choice of inner 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, in differential geometry each tangent space to a manifold may be endowed with an inner product, giving rise to what is called a Riemannian 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 that tangent spaces 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 called Hilbert spaces 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 an orthogonal matrix and a symmetric positive definite matrix, which is called a polar decomposition. Singular matrices can be also factored, but not uniquely.

A symmetric ntimes n matrix is determined by tfrac{n(n+1)}{2} scalars. Similarly, a skew-symmetric matrix is determined by tfrac{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 Rn can be uniquely written in the form q(x) = xTAx 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 Rn, "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.

See also

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

See also symmetry in mathematics.

References

  • A. J. Bosch (1986). "The factorization of a square matrix into two symmetric matrices". American Mathematical Monthly 93 462–464.

External links

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