Symmetric matrix

In linear algebra, a symmetric matrix is a matrix that is its own transpose. Thus A is symmetric if

A^\textrm{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 spectral theorem, which says that any symmetric matrix whose entries are real can be diagonalized by an orthogonal matrix. More explicitly: to every symmetric real matrix A there exists a real orthogonal matrix Q such that D = Q^TAQ is a diagonal matrix. Every symmetric matrix is thus, up to choice of an orthonormal basis, a diagonal matrix.

Another way of stating the spectral theorem is that the eigenvectors 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=\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. Two real symmetric matrices commute if and only if they have the same eigenspaces.

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 <,> 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 \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 Rⁿ can be uniquely written in the form q(x) = x^TAx 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 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.

References

Matrices

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Properties

Occurrence

See also

References